Arithmetic and geometric sequences - Hawker Maths 2022

ChapTer 5 • Arithmetic and geometric sequences 171

ChapTer ConTenTS 5a Recognition of arithmetic sequences 5B Finding the terms of an arithmetic sequence 5C The sum of a given number of terms of an arithmetic sequence 5d Recognition of geometric sequences 5e Finding the terms of a geometric sequence 5F The sum of a given number of terms of a geometric sequence 5G Applications of geometric sequences 5h Finding the sum of an infinite geometric sequence 5i Contrasting arithmetic and geometric sequences through graphs

diGiTal doC10 Quick Questionsdoc-9435

ChapTer 5

Arithmetic and geometric sequences

introductionPatterns occur naturally in many real-life situations; for example the addition of interest to bank accounts, plant spacing in a winery and the stacking of logs in a pile. Two of the most common patterns are termed arithmetic and geometric sequences. Recognition of these two patterns is important in analysing situations that occur normally in the real world. Look at the sequence in the shaded column on this bank statement.

Date Description Debit Credit Balance

1.1.2006 Deposit 1000.00 1000.00

1.1.2007 Interest 100.00 1100.00

1.1.2008 Interest 100.00 1200.00

1.1.2009 Interest 100.00 1300.00

5a recognition of arithmetic sequencesA sequence in mathematics is an ordered set of numbers.

An arithmetic sequence is one in which:1. the difference between any two successive terms is the same, or2. the next term in the sequence is found by adding the same number.

Consider the arithmetic sequence:4, 7, 10, 13, 16, 19, 22.

The difference between each successive term is +3, or similarly, the next term is found by adding 3 to the previous term. We can see that a positive common difference gives a sequence that is increasing. We say that the common difference is +3, stated as d = +3.

4 7

+310

+313

+316

+319

+322

+3

The fi rst term of the sequence is 4. We refer to the fi rst term of a sequence as ‘a’. So in this example, a = 4.

Concept summary

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Contents

inTrodUCTion 171

5a recognition of arithmetic sequences 171 exercise 5a recognition of arithmetic sequences 173

5B Finding the terms of an arithmetic sequence 175 exercise 5B Finding the terms of an arithmetic sequence 177

5C The sum of a given number of terms of an arithmetic sequence 178 exercise 5C The sum of a given number of terms of an arithmetic sequence 181

5d recognition of geometric sequences 183 exercise 5d recognition of geometric sequences 185

172 Maths Quest 12 Further Mathematics

In this arithmetic sequence, the first term is 4, the second term is 7, the third term is 10 and so on. Another way of writing this is:

t1 = 4, t2 = 7 and t3 = 10.

There are 7 terms in this sequence. Because there is a countable number of terms in the sequence, it is referred to as a finite sequence.

The arithmetic sequence below

37, 30,

−723,

−716,

−79 ...

−7

is an infinite sequence since it continues endlessly as indicated by the ellipsis (. . .) after the final term shown. The first term, a, is 37 and the common difference, d, is −7. We can see that a negative common difference gives a sequence that is decreasing.

1. An arithmetic sequence is a sequence of numbers for which the difference between successive terms is the same.

2. The first term of an arithmetic sequence is referred to as ‘a’.3. The common difference between successive terms is referred to as ‘d’.4. tn is the term number; for example, t6 refers to the 6th term in the sequence.

Worked example 1

Which of the following are arithmetic sequences?a 7, 13, 19, 25, 31, . . . b −81, −94, −106, −120, −133, . . .c 1.3, 2.5, 3.7, 4.9, 6.3, . . . d −11

2, −1,

−12

, 0, 12, . . .

Think WriTe

a 1 Write the sequence. a 7, 13, 19, 25, 31, …

2 Calculate the difference between the first term, t1, and the second term, t2.

t2 − t1 = 13 − 7 = 6

3 Calculate the difference between the second term, t2, and the third term, t3.

t3 − t2 = 19 − 13 = 6

4 Calculate the difference between the third term, t3, and the fourth term, t4.

t4 − t3 = 25 − 19 = 6

5 Calculate the difference between t5 and t4. t5 − t4 = 31 − 25 = 6

6 Check that the differences are the same and write your answer.

There is a common difference of 6, therefore d = 6.This is an arithmetic sequence.

b 1 Write the sequence. b −81, −94, −106, −120, −133, . . .

2 Calculate the difference between the first term, t1, and the second term, t2.

t2 − t1 = −94 − (−81) = −13

3 Calculate the difference between the second term, t2, and the third term, t3.

t3 − t2 = −106 − (−94) = −12

4 There is no need to do any further checks as the two differences are not the same.

There is no common difference.This is not an arithmetic sequence.

c 1 Write the sequence. c 1.3, 2.5, 3.7, 4.9, 6.3, . . .

2 Calculate the difference between t2 and t1. 2.5 − 1.3 = 1.2





6 Check that the differences are the same. There is no common difference.This is not an arithmetic sequence.

d 1 Write the sequence. d −112, −1,

−12, 0, 1

2, …

2 Calculate the difference between t2 and t1. −1 − −112 = 1

2

3 Calculate the difference between t3 and t2.−1

2 − −1 = 1

2

4 Calculate the difference between t4 and t3. 0 − −1

2 = 1

2

5 Calculate the difference between t5 and t4.12 − 0 = 1

2

6 Check that the differences are the same. There is a common difference of 12.

This is an arithmetic sequence.

Worked example 2

Write the value of a and d for each of the following arithmetic sequences.a 1.2, 3.6, 6, 8.4, 10.8, . . . b −12

5,

−25, 3

5, 13

5, 2 3

5, . . .

Think WriTe

a 1 Write the sequence. a 1.2, 3.6, 6, 8.4, 10.8, …

2 What is the fi rst term? a = 1.2

3 What is the difference between t2 and t1?You need check only once as the question states that this is an arithmetic sequence.

t2 − t1 = 3.6 − 1.2= +2.4

d = +2.4

4 Write your answer. The arithmetic sequence has a fi rst term, a, of 1.2 and a common difference, d, of +2.4.

b 1 Write the sequence. b −125,

−25, 3

5, 13

5, 23

5, . . .

2 What is the fi rst term? a = −125

3 What is the difference between t2 and t1? t2 − t1 = −2

5 − −12

5

= +1 d = +1

4 Write your answer. The arithmetic sequence has a fi rst term, a,

of −125 and a common difference, d, of +1.

exercise 5a recognition of arithmetic sequences 1 We1a State which of the following are arithmetic sequences.

a 2, 7, 12, 17, 22, . . . b 2, 4, 8, 16, 32, . . . c 2, 4, 6, 8, 10, . . .d 3, 7, 11, 15, 20, . . . e 4, 8, 11, 15, 19, . . . f 3, 30, 300, 3000, 30 000, . . .g 1, 1, 2, 4, 8, . . . h 2, 2, 4, 4, 8, . . . i 10, 20, 30, 40, 50, . . .

2 We2 For those arithmetic sequences found in question 1, write the values of a and d.


3 We1b State which of the following are arithmetic sequences.

a −123, −23, 77, 177, 277, . . . b −1, 3, 1, 5, 3, . . .c −7, −1, 5, 11, 17, . . . d −67, −27, 13, 53, 93, . . .e 5, −2, −9, −16, −23, . . . f −7, −18, −29, −30, −39, . . .g 1, 0, −1, −3, −5, . . . h 0, −10, −21, −32, −43, . . .

4 For those arithmetic sequences found in question 3, write the values of a and d.

5 We1c State which of the following are arithmetic sequences.

a 0.7, 1, 1.3, 1.6, 1.9, . . . b 2.3, 3.2, 5.2, 6.2, 7.2, . . .c −3.5, −2, −0.5, 1, 2.5, . . . d −2.7, −2.5, −1.7, −1.5, −0.7, . . .e 2, 0.1, −2.1, −3.3, −4.5, . . . f −5.2, −6, −6.8, −7.6, −8.4, . . .


7 We1d State which of the following are arithmetic sequences.

a 12, 11

2, 21

2, 31

2, 41

2, . . . b 1

4, 3

4, 11

4, 13

4, 21

4, . . .

c 15, 3

5, 1, 12

5, 14

5, . . . d

−34

, 0, 34, 11

2, 21

4, . . .

e 13,

−13

, −1, −113, 2, . . . f

−12

, −14

, −16

, −18

, −110

, . . .


9 Show which of the following situations are arithmetic sequences.a A teacher hands out 2 lollies to the fi rst

student, 4 lollies to the second student, 6 lollies to the third student and 8 lollies to the fourth student.

b The sequence of numbers after rolling a die 8 times.

c The number of layers of paper after each folding in half of a large sheet of paper.

d The house numbers on the same side of a street on a newspaper delivery route.e The cumulative total of the number of seats in the fi rst ten rows in a regular cinema (for example,

with 8 seats in each row, so there are 8 seats after the fi rst row, 16 seats after the fi rst 2 rows, and so on).

10 For those arithmetic sequences found in question 9, where appropriate information is given, write the value of a and d.

11 For the following arithmetic sequences:a 4, 13, 22, 31, . . . which term, tn, will be equal to 58?b 9, 4.5, 0, . . . which term, tn, will be equal to −18? c −60, −49, −38, . . . which term, tn, will be the fi rst to be greater than 10? d 100, 87, 74, . . . which term, tn, will be the fi rst to be less than 58?

12 Jenny receives 5 dollars for completing the first kilometre of a walkathon and 7 dollars more for completing each subsequent kilometre. Write the arithmetic sequence that represents the amount received by Jenny for each kilometre walked from 1 to 10 kilometres.

13 Each week, Johnny buys a pack of 9 basketball cards. In the first week Johnny has 212 cards in his collection. Give the total number of cards Johnny has for each of the first five weeks.

14 mC Which of the following could be the first five terms of an arithmetic sequence?

a 1, 3, 9, 12, 15, . . . B −266, −176, −86, 4, 94, . . .C 3, 3, 6, 6, 9, . . . d 0, 1, 2, 4, 8, . . .e −3, −1, 0, 1, 3, . . .

15 mC 3, 10, 17, 24, 31, 38For the arithmetic sequence above, it is true to say that it is:a an infi nite sequence with a = 3 and d = 7 B an infi nite sequence with a = 7 and d = 3C an infi nite sequence with t2 = 10 and d = 7 d a fi nite sequence with a = 3 and d = 7e a fi nite sequence with a = 7 and d = 3


5B Finding the terms of an arithmetic sequenceConsider the arithmetic sequence for which a = 8 and d = 10.

8 18 28 38 48

+10 +10 +10 +10

Now, t1 = 8 t1 = at2 = 8 + 10 t2 = a + d t2 = a + 1dt3 = 8 + 10 + 10 t3 = a + d + d t3 = a + 2dt4 = 8 + 10 + 10 + 10 t4 = a + d + d + d t4 = a + 3dt5 = 8 + 10 + 10 + 10 + 10 t5 = a + d + d + d + d t5 = a + 4d

We notice a pattern emerging. That pattern can be described by the equation:

tn = 8 + (n − 1) × 10

where n represents the number of the term.For example, if n = 4, then the fourth term is: t4 = 8 + (4 − 1) × 10 = 8 + 3 × 10 = 38.Therefore, the 4th term is 38.

We can generalise this rule for all arithmetic sequences.

tn = a + (n − 1) dwhere tn is the nth term a is the fi rst term d is the common difference.

This rule enables us to fi nd any term of an arithmetic sequence provided we know the value of a and d.

Worked example 3

Find the 20th term of the following arithmetic sequence.5, 40, 75, 110, 145, . . .

Think WriTe

1 Find the value of a. a = 5

2 Find the value of d. You need to calculate only one difference as the question states that it is an arithmetic sequence.

d = t2 − t1= 40 − 5= 35

3 Use the rule tn = a + (n − 1) d where n is 20 for the 20th term.

t20 = 5 + (20 − 1) × 35= 5 + 19 × 35= 670

4 Write the answer. The 20th term is 670.

If we are given only two terms of an arithmetic sequence, we are able to use the rule tn = a + (n − 1) d to set up two simultaneous equations to fi nd the value of a and d and hence write the rule for the arithmetic sequence.

Worked example 4

The third term of an arithmetic sequence is −1 and the fi fth term is 11.a Write the rule for the arithmetic sequence. b Find the 50th term of the sequence.

Think WriTe

a 1 We know that t3 = −1 and that tn = a + (n − 1) d, where n = 3.

a t3 = a + 2d = −1

inTeraCTiViTYint-0007number patterns


2 We know that t5 = 11 and that tn = a + (n − 1) d, where n = 5.

t5 = a + 4d = 11

3 Solve the 2 equations simultaneously using the elimination technique.Eliminate a, by subtracting equation [1] from equation [2].

a + 2d = −1 [1] a + 4d = 11 [2] 2d = 12 [2] − [1] d = 6

4 Evaluate a by substituting d = 6 into either of the two equations.

Substituting d = 6 into [1]:a + 12 = −1 a = −13

5 To fi nd the rule, substitute values for a and d into tn = a + (n − 1) d.

tn = −13 + (n − 1) × 6 = −13 + 6n − 6 = −19 + 6n

b 1 To fi nd the 50th term or t50, substitute n = 50 into the rule found.

b tn = −19 + 6nt50 = −19 + 6 × 50 = −19 + 300 = 281

2 Write your answer. The 50th term is 281.

Worked example 5

If the fi rst three terms of an arithmetic sequence are 5.2, 7.4 and 9.6, which term is equal to 53.6?

Think WriTe

1 Find the rule for the arithmetic sequence.• Identify the value of a (it is the 1st term).• Calculate the value of d.• Substitute the values of a and d into

tn = a + (n − 1) d.

a = 5.2 d = t2 − t1 = 7.4 − 5.2 = 2.2tn = 5.2 + (n − 1) × 2.2 = 5.2 + 2.2n − 2.2 = 3 + 2.2n

2 Which term is equal to 53.6?Substitute 53.6 for tn and solve for n.

53.6 = 3 + 2.2n

n = 50.6

2.2 = 23

3 Write your answer. The term 53.6 is the 23rd term.

Worked example 6

An ant colony is studied and found to have a population of 10 000 in the fi rst week of the study. The population increases by 500 each week after that.a Write a rule for the number of ants in the colony in week n of the study.b When will the ant population double in size?

TUTorialeles-1268Worked example 5


Think WriTe

a 1 We know that a = 10 000 and d = 500 and that tn = a + (n − 1) d.

a tn = 10 000 + (n − 1) × 500= 10 000 + 500n − 500

2 Substitute and simplify. tn = 9500 + 500n

b 1 Using the rule found, we need to find which term is equal to 10 000 doubled or tn = 20 000.

b tn = 9500 + 500n20 000 = 9500 + 500n10 500 = 500n

n = 10 500

500 = 21

2 Write your answer. The ant population will double to 20 000 in the 21st week.

exercise 5B Finding the terms of an arithmetic sequence

1 We3 For each of the arithmetic sequences given, find:a the 25th term of the sequence 2, 7, 12, 17, 22, . . .b the 19th term of the sequence 2, 4, 6, 8, 10, . . .c the 30th term of the sequence 0, 100, 200, 300, 400, . . .d the 27th term of the sequence −7, −1, 5, 11, 17, . . .e the 33rd term of the sequence 5, −2, −9, −16, −23, . . .f the 39th term of the sequence 14.1, 28.2, 42.3, 56.4, 70.5, . . .

2 We4 Evaluate the following.a The 2nd term of an arithmetic sequence is 13 and the 5th term is 31. What is the 17th term of

this sequence?b The 2nd term of an arithmetic sequence is −23 and the 5th term is 277. What is the 20th term of

this sequence?c The 2nd term of an arithmetic sequence is 0 and the 6th term is −8. What is the 32nd term of this

sequence?d The 3rd term of an arithmetic sequence is 5 and the 7th term is −19. What is the 40th term of this

sequence?

3 We5 Evaluate the following.a The first 3 terms of an arithmetic sequence are 3, 9 and 15. Which term is equal to 141?b The first 3 terms of an arithmetic sequence are −9, −6 and −3. Which term is equal to 72?c The first 3 terms of an arithmetic sequence are 1.7, 2.5 and 3.3. Which term is equal to 28.1?d The first 3 terms of an arithmetic sequence are −1.5, −2 and −2.5. Which term is equal to −140.5?

4 We6 A batsman made 23 runs in his first innings, 33 in his second and 43 in his third. If he continued to add 10 runs each innings, write a rule for the number of runs he would have made in his nth innings.

5 In a vineyard, rows of wire fences are built to support the vines. The length of the fence in row 1 is 40 m, the length of the fence in row 2 is 43 m, and the length of the fence in row 3 is 46 m. If the lengths of the fences continue in this pattern, write down a rule for the length of a fence in row number n.


6 A marker is placed 15 m from a white line by a P.E. teacher. The next marker is placed 25 m from the white line and the next 35 m from the white line. The teacher continues placing markers in this pattern.a Write a rule for the distance of marker n from the white line.b How many markers will need to be placed before the last marker is at least 100 metres from

the line?

7 mC The 41st term of the arithmetic sequence −4.3, −2.1, 0.1, 2.3, 4.5, . . . is:a 83.7 B 85.9 C 92.3 d 172.4 e 178.5

8 mC The 2nd term of an arithmetic sequence is −2 and the 5th term is 2.5. The 27th term of this sequence is:a 32.5 B 35.5 C 42.5 d 89.5 e 96

9 mC The numbers 8, 1 and −6 form the first three terms of an arithmetic sequence. In this arithmetic sequence the term which is equal to −258 is the:a 30th B 32nd C 37th d 39th e 42nd

10 Find the 28th term of the arithmetic sequence −5.2, −6, −6.8, −7.6, −8.4, . . . 11 Find the 31st term of the arithmetic sequence 1

5, 35, 1, 12

5 , 145 , . . .

12 The 3rd term of an arithmetic sequence is −16 and the 5th term is −4.2. What is the 19th term of this sequence?

13 The 4th term of an arithmetic sequence is 312 and the 7th term is 6

12. What is the 25th term of this

sequence?

14 The 3rd term of an arithmetic sequence is 15 and the 8th term is 45. Which term of the sequence is equal to 183?

15 The 2nd term of an arithmetic sequence is 1 and the 6th term is −15. Which term of the sequence is equal to −167?

16 mC The 3rd term of a sequence is −1 and the fifth is 14. The term which is equal to 141.5 is the:a 9th B 11th C 18th d 20th e 22nd

17 Peter plants his first tomato seedling 0.5 m from the fence, the next 1.3 m from the fence and the next 2.1 m from the fence. If he continues to plant in this pattern, how far will the 14th seedling be from the fence?

18 Olivia began her china collection in 1951. She was given 3 pieces of china that year and added 2 pieces each year after that. How many pieces did Olivia have in her collection in the year 2000?

19 The membership of a local photography club was 7 in its first year. If the club added 4 members to its membership each year, write a rule for the number of members in the club in year n.

20 The first fence post in a fence is 12 m from the road, the next is 15.5 m from the road and the next is 19 m from the road. The remainder of the fence posts are spaced in this pattern.a Write a rule for the distance of fence post n from the road.b If 100 posts are to be erected, how far will the last post be from the road?

5C The sum of a given number of terms of an arithmetic sequenceWhen the terms of an arithmetic sequence are added together, an arithmetic series is formed. So, 5, 9, 13, 17, 21, . . . is an arithmetic sequence whereas 5 + 9 + 13 + 17 + 21 + . . . is an arithmetic series.

The sum of n terms of an arithmetic sequence is given by Sn.Consider the finite arithmetic sequence below.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10


The sum of this arithmetic sequence is given by S10 since there are 10 terms in the sequence.So, S10 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

= 55Note that the sum of the fi rst and last terms is 11. Also, the sum of the second and second last

terms is 11. Similarly, the sum of the third and third last term is 11. This pattern continues with the fourth and fourth last terms as well as with the fi fth and fi fth last terms. There are in fact fi ve lots of 11.

We can formalise this pattern to obtain a rule which applies to all arithmetic sequences.Let Sn = a + (a + d ) + (a + 2d ) + . . . + (l − 2d ) + (l − d ) + lwhere l is the last term of the sequence.

By reversing the order of the series above, we obtain

Sn = l + (l − d ) + (l − 2d ) + . . . + (a + 2d ) + (a + d ) + a

By adding these two equations, we obtain

2Sn = (a + l) + (a + d + l − d ) + (a + 2d + l − 2d ) + . . . (l − d) + (a + d) + (a + l)2Sn = (a + l) + (a + l) + (a + l) + . . . (a + l) + (a + l)2Sn = n(a + l) where n represents the number of terms in the sequence.

So, Sn = 12n(a + l)

The sum of n terms of an arithmetic sequence with a as its fi rst term and l as its last term is given by:

Sn = n2

(a + l).

Recall that the nth term of an arithmetic sequence is given by:

tn = a + (n − 1)d.

So, for the sum of n terms, l is the last term; that is, tn = l.So, the last term is:

l = a + (n − 1)d.

Substituting this into Sn = n

2(a + l )

we obtain Sn = n

2{a + [a + (n − 1)d]}

= n

2[2a + (n − 1)d]

An alternative formula for the sum of n terms of an arithmetic sequence when the value of a and d are known, is given by:

Sn = n2

[2a + (n − 1)d ].

Worked example 7

Find the sum of the fi rst ten given terms of the arithmetic sequence:4, 10, 16, 22, 28, 34, 40, 46, 52, 58.

Think WriTe

Method 1:

1 We know the values of the fi rst and last term and that there are ten terms in the series.

a = 4l = 58n = 10

2 Use the series formula Sn = n

2(a + l). Sn =

n

2(a + l)

S10 = 102

(4 + 58)

= 5 × 62 = 310


Method 2: 1 We know the value of a and d and n. a = 4

d = 10 − 4 = 6n = 10

2 Use the formula Sn = n

2[2a + (n − 1)d]. Sn =

n

2[2a + (n − 1)d]

Sn = 102

[2 × 4 + (10 − 1)6]

S10 = 5[8 + 9 × 6] = 5[8 + 54] = 5 × 62 = 310

3 Write the answer. The sum of the fi rst 10 terms is 310.

Worked example 8

The fi rst term of an arithmetic sequence is 5 and the seventh term is 29. What is the sum of the fi rst 10 terms of this sequence?

Think WriTe

1 Find the value of a and d.The value of a is given.To fi nd the common difference, d, use the formula tn = a + (n − 1)d, as we know the value of the 7th term is 29.

a = 5 tn = a + (n − 1)d t7 = 5 + 6 × d = 295 + 6d = 29 6d = 24 d = 4


2[2a + (n − 1)d]. S10 =

102

[2 × 5 + (10 – 1)4]

= 5[10 + 9 × 4] = 230

3 Write your answer. If a = 5 and d = 4, the sum of the fi rst ten terms is 230.

Worked example 9

The 3rd term of an arithmetic sequence is 4 and the 8th is −11. What is the sum of the fi rst 30 terms of the sequence?

Think WriTe

1 Find out the value of a and d by setting up two simultaneous equations using tn = a + (n − 1)d.

t3 = a + 2d = 4 [1] t8 = a + 7d = −11 [2]

2 Eliminate a by subtracting equation 1 from equation 2.

a + 2d = 4 [1] a + 7d = −11 [2] 5d = −15 [2] – [1] d = −3 Substitute d = −3 into equation [1]. a + 2d = 4

a + 2 × −3 = 4 a – 6 = 4 a = 10




2[2a + (n − 1)d ].

To fi nd the sum of the fi rst 30 terms.

S30 = 302

[2 × 10 + (30 – 1)(−3)]

= 15[20 + 29 × −3]= −1005

4 Write your answer. If a = 10 and d = −3, the sum of the fi rst 30 terms is −1005.

Worked example 10

The fi rst term of a sequence is −7 and the sum of the fi rst 25 terms is 1625. Find:a the 25th termb the fi rst fi ve terms of the sequence.

Think WriTe

a 1 We know a = −7 and S25 = 1625 and the 25th term is the last term, l.

Use the formula Sn = n

2(a + l ).

a S25 = 252

(−7 + l )

= 162512.5(−7 + l ) = 1625

−7 + l = 162512.5

−7 + l = 130l = 137

2 Write your answer. l = t25 = 137

The 25th term is 137.

b 1 To fi nd the common difference, d, we can

use Sn = n

2[2a + (n − 1)d ] or tn = a + (n − 1)d

having found that the 25th term is 137.

b S25 = 252

[−14 + (25 − 1)d ]

= 162512.5 [−14 + 24d ] = 1625

−14 + 24d = 130 24d = 144

d = +6or tn = a + (n − 1)d

t25 = 137 137 = −7 + (25 − 1)d 144 = 24d

d = +6

2 We know that a = −7 and d = +6, so generate the fi rst fi ve terms of the sequence.

The sequence is −7, −1, 5, 11, 17, . . .

exercise 5C The sum of a given number of terms of an arithmetic sequence 1 We 7 For each of the given series, find:

a the sum of the fi rst 20 terms of the sequence 3, 7, 11, 15, 19, . . .b the sum of the fi rst 34 terms of the sequence 10, 20, 30, 40, 50, . . .c the sum of the fi rst 23 terms of the sequence −4, −1, 2, 5, 8, . . .d the sum of the fi rst 29 terms of the sequence 10, 7, 4, 1, −2, . . .

2 We8 The first term of an arithmetic sequence is 5 and the second is 9. Find the sum of the first 40 terms of the sequence.

3 The first term of an arithmetic sequence is 0.7 and the second is 1. Find the sum of the first 25 terms of the sequence.


4 We9 For each of the following, evaluate the sum of a series, Sn.a The 3rd term of an arithmetic sequence is 19 and the 4th is 25. Find the sum of the fi rst 15 terms

of the sequence.b The 2nd term of an arithmetic sequence is 3.6 and the 5th is 10.8. Find the sum of the fi rst

23 terms of the sequence.c The 3rd term of an arithmetic sequence is −0.5 and the 6th is 4. Find the sum of the fi rst 26 terms

of the sequence.d The 2nd term of an arithmetic sequence is 0.75 and the 5th is 2.25. Find the sum of the fi rst

27 terms of the sequence.

5 We 10 The first term of an arithmetic sequence is 2 and the sum of the first 19 terms of the sequence is 551. Find:a the 19th term b the fi rst 3 terms of the sequence.

6 The first term of an arithmetic sequence is −4 and the sum of the first 30 terms of the sequence is 2490. Find:a the 30th term b the fi rst 3 terms of the sequence.

7 mC The sum of the first 21 terms of the sequence, 0, −312, −7, −10 1

2, −14, . . . is:

a −1470 B −735 C −700 d 36.75 e 735

8 mC The first term of an arithmetic sequence is −5.2 and the second is −6. The sum of the first 22 terms of the sequence is:

a −598.4 B −299.2 C −242 d −70.4 e 70.4

9 What is the sum of the first 19 terms of the sequence −180, −80, 20, 120, 220, . . . ? 10 What is the sum of the first 28 terms of the sequence

−12, 1

2, 11

2, 21

2, . . . ?

11 The 2nd term of an arithmetic sequence is 28.2 and the 6th is 84.6. Find the sum of the first 40 terms of the sequence.

12 The 1st term of an arithmetic sequence is 5.5 and the sum of the first 18 terms of this sequence is 328.5. Find:a the 18th term b the fi rst 3 terms of the sequence.

13 The 1st term of an arithmetic sequence is 11 and the sum of the first 20 terms of this sequence is −350. Find:a t20 b the fi rst 3 terms of the sequence.

14 Sam makes $100 profit in his first week of business. If his profit increases by $75 each week, what would his total profit be by the end of week 15?

15 George’s salary is to start at $36 000 a year and increase by $1200 each year after that. How much will George have earned in total after 10 years?

16 A staircase is designed so that the height of each step increases by 0.8 cm for each step. If the height of the first step is 15 cm, what is the total height of the first 17 steps?

17 Paula collects stamps. She bought 250 in the first month to start her collection and added 15 stamps to the collection each month thereafter. How many stamps will she have collected after 5 years?

18 Proceeds from the church fete were $3000 in 1981. In 1982 the proceeds were $3400 and in 1983 they were $3800. If they continued in this pattern:a what would be the proceeds from the year

2000 fete?b how much in total would the proceeds from

church fetes from 1981 to 2000 have amounted to?

19 Fees for groups meeting at the community centre rise by $5 each year. If the fees started at $60 a year in the first year:a how much will the fees be in the 20th year?b how much would a group which had met at the centre for all of those 20 years have paid in total?

diGiTal doCdoc-9436

WorkSHEET 5.1


5d recognition of geometric sequencesA sequence in mathematics is an ordered set of numbers. A geometric sequence is one in which the fi rst term is multiplied by a number, known as the common ratio, to create the second term which is multiplied by the common ratio to create the third term, and so on. The fi rst term in a geometric sequence is referred to as a and the common ratio is referred to as r.

Consider the geometric sequence where a = 1 and r = 3. The terms in the sequence are:

1 3×3

9 27 81...×3 ×3 ×3

To discover the common ratio, r, of a geometric sequence you need to calculate the ratio of any

two successive terms, for example, t2t1

. You could alternatively calculate t3t2

or t4t3

and so on.

A geometric sequence is a sequence of numbers for which the ratio of successive terms is the same.

t2t1

= t3t2

= t4t3

= . . . = common ratio

The fi rst term of a geometric sequence is referred to as a.The common ratio between a term and its preceding term is referred to as r.

Worked example 11

Which of the following are geometric sequences? For those that are geometric, state the values of a and r.a 2, 10, 50, 250, 1250, . . . b 4, −8, 16, −32, 64, . . .c −2, −6, 18, 54, −162, . . . d 6, 3, 3

2, 3

4, 3

8, . . .

Think WriTe

a 1 Write the sequence. a 2, 10, 50, 250, 1250, . . .

2 Calculate the ratio of t2t1

.t2t1

= 10

2 = 5


.t3t2

= 50

10 = 5


.t4t3

= 250

50 = 5


.t5t4

= 1250

250 = 5

6 Check that all ratios are the same. There is a common ratio of 5.This is a geometric sequence where a = 2 and r = 5.

b 1 Write the sequence. b 4, −8, 16, −32, 64, . . .


. t2t1

= −8

4 = −2


. t3t2

= 16−8

= −2

Concept summary


Units: 3 & 4

AOS: 1

Topic: 1

Concept: 3

Do more Interact

with geometric sequences.



. t4t3

= −32

16 = −2


. t5t4

= 64

−32 = −2

6 Check that all ratios are the same. There is a common ratio of −2.This is a geometric sequence where a = 4 and r = −2.

c 1 Write the sequence. c −2, −6, 18, 54, −162, . . .


. t2t1

= −6−2

= +3


. t3t2

= 18−6

= −3

4 There is no need to do any further check as the two ratios are not the same.

There is no common ratio.This is not a geometric sequence.

d 1 Write the sequence. d 6, 3, 3

2, 3

4, 3

8, . . .

2 Calculate t2t1

.t2t1

= 3

6

= 12

3 Calculate t3t2

.t3t2

= 3

6 ÷ 3

= 32 ×

1

3

= 12

4 Calculate t4t3

.t4t3

= 3

4 ÷

3

2

= 34 ×

2

3

= 24

= 12

5 Calculate t5t3

.t5t4

= 3

8 ÷

3

4

= 38 ×

4

3

= 12

6 Check that all ratios are the same. There is a common ratio of 12.

This is a geometric sequence where a = 6

and r = 12.


exercise 5d recognition of geometric sequences1 We 11 State which of the following are geometric sequences.

a 1, 2, 4, 8, 16, . . . b 2, 6, 18, 54, 162, . . .c 1, 4, 16, 64, 256, . . . d 2, 6, 12, 24, 48, . . .e 1, 5, 25, 100, 125, . . . f 3, 6, 12, 24, 48, . . .g 5, 10, 20, 40, 80, . . . h 2, 3, 4, 5, 6, . . .

2 Look again at the geometric sequences found in question 1. Write the values of a and r.

3 State which of the following sequences are geometric sequences.a −1, −2, −4, −8, −16, . . . b −2, −4, 8, 16, −32, . . .c 0, −3, −9, −27, −81, . . . d −2, −6, −18, −54, −162, . . .e −4, −8, −16, −32, −64, . . . f −9, −3, 1, 3, 9, . . .g 1, 5, −25, 125, −625, . . . h −5, −20, −80, −320, −1280, . . .

4 For those geometric sequences found in question 3, write the value of a and r.

5 State which of the following sequences are geometric sequences.a 1, −2, 4, −8, 16, . . . b 1, 3, −9, 27, −81, . . .c 4, −12, 36, −108, 324, . . . d 7, −7, 7, −7, 7, . . .e 3, −6, 18, −54, 112, . . . f −2, 8, −12, 24, −48, . . .


7 State which of the following sequences are geometric sequences.a −2, 4, −8, 16, −32, . . . b −1, −5, 10, −20, −40, . . .c −3, −15, −75, −375, −1875, . . . d −10, −1, 0, 1, 10, . . .e 0, −4, −16, −64, −256, . . . f −6, 60, −600, 6000, −60 000, . . .


9 State which of the following sequences are geometric sequences.a 1.2, 2.4, 4.8, 9.6, 19.2, . . . b 2.3, 6.9, 13.8, 27.6, 55.2, . . .c 2.25, 4.5, 9, 18, 36, . . . d 7, 3.5, 1.75, 0.875, 0.4375, . . .e 10, 12, 14.4, 17.28, 20.736, . . . f −1.2, 3.6, 10.8, 21.6, 43.2, . . .


11 State which of the following sequences are geometric sequences.

a 12, 1

4, 1

8, 1

16, 1

32, . . . b 1

2, 1

6, 1

12, 1

18, 1

24, . . . c 1

4, 1

20, 1

100, 1

500, 1

2500, . . .

d 13, 2

3, 11

3, 22

3, 51

3, . . . e 1

5, 1

10, 1

15, 1

20, 1

25, . . . f 1, 1

3

−

, 1

9, 1

12

−

, 1

81, . . .


13 mC Which of the following is a geometric sequence?

a 1, 3, 6, 9, 12, . . . B 0.002, 0.02, 0.2, 20, 200, . . .C 9, −9, 3, −3, 1, . . . d 5, −5, 5, −5, 5, . . .e 1

8, 1

7, 1

6, 1

6, 1

5, 1

4, . . .

14 mC There is a geometric sequence for which a is positive and r = −2. It is true to say that:a only one term of the sequence is a positive numberB the third term will be a negative numberC the third term will be less than the second termd the 5th term would be greater than the 6th terme the 4th term would be greater than the third term

15 mC There is a geometric sequence for which every term is negative. It could be said with certainty that:a a and r are both positive B a and r are both negativeC a is positive and r is negative d a is negative and r is positivee a is greater than r

16 mC There is a geometric sequence for which every odd-numbered term is positive and every even-numbered term is negative. It could be said with certainty that:a a and r are both positive B a and r are both negativeC a is positive and r is negative d a is negative and r is positivee a is greater than r


17 A savings account balance at the end of each of the past four years is given as follows: $100.00, $110.00, $121.00, $133.10.a Prove this is a geometric sequence.b State the value of a and r.

18 On the first day Jenny hears a rumour. On the second day, she tells two friends. On the third day, each of these two friends tell two of their own friends, and so on.a Write the geometric sequence for the fi rst fi ve days of this real-life situation.b Find the value of r.c How many people are told of the rumour on the 12th day?

19 Decay of radioactive material is modelled as a geometric sequence where r = 1

2. If there are 20 million radioactive atoms,

write the first 7 terms of the sequence.

20 Copy and complete the following.a In a sequence that grows and where each term is

positive, the r value is than 1.b In a sequence that decays and where each term is

positive, the r value is between and .

c In a sequence like 2, 2, 2, 2, 2, . . . , the r value is 1.

d In a sequence like 2, −6, 18, −54, . . ., the r value is less than .

e In a sequence like −54, 18, −6, 2, −1, . . . , the r value is between and .

5e Finding the terms of a geometric sequenceConsider the fi nite geometric sequence of seven terms for which a = 3 and r = 4.

3 12×4

48 192 384×4 ×4 ×4

Now, t1 = 3 t1 = a t2 = 3 × 4 t2 = a × r t2 = a × r1

t3 = 3 × 4 × 4 t3 = a × r × r t3 = a × r 2

t4 = 3 × 4 × 4 × 4 t4 = a × r × r × r t4 = a × r 3

t5 = 3 × 4 × 4 × 4 × 4 t5 = a × r × r × r × r t5 = a × r 4 and so on . . .


We notice a pattern emerging. That pattern can be described by the equation:

tn = 3 × 4n − 1.

For example, if n = 5,

t5 = 3 × 44.

We can generalise this rule for all geometric sequences.

tn = ar n − 1

where tn is the nth term a is the fi rst term r is the common ratio.

This rule enables us to fi nd any term of a geometric sequence provided we know the value of a and r.

Worked example 12

Find the 12th term of the geometric sequence:2, 10, 50, 250, 1250, . . .

Think WriTe

1 State the value of a. a = 2

2 It has been stated that it is a geometric sequence, so fi nd the value of r.

r = 102

= 5

3 Use the rule tn = a × r n − 1 to fi nd the 12th term. t12 = 2 × 512 − 1

= 97 656 250

4 Write the answer. The value of the 12th term is 97 656 250.

Worked example 13

The 2nd term of a geometric sequence is 8 and the 5th is 512. Find the 10th term of this sequence.

Think WriTe

1 We know that t2 = 8 and that tn = a × r n − 1. t2 = a × r1

= 8

2 We know that t5 = 512 and that tn = a × r n − 1. t5 = a × r4

= 512

3 Solve the 2 equations simultaneously by eliminating a, to fi nd r.Divide equation 2 by equation 1.

a × r1 = 8 [1] a × r4 = 512 [2]a × r4

a × r=

512

8 [2] ÷ [1]

r3 = 64r = 4

4 To fi nd a, substitute the value of r. Substitute r = 4 into equation [1].a × 4 = 8

a = 2

5 Write the rule. tn = 2 × 4n − 1

6 To fi nd the 10th term, let n = 10. t10 = 2 × 49

= 524 288

7 Write your answer. The 10th term in the sequence is 524 288.


Worked example 14

The fi rst three terms of a geometric sequence are 2, 6 and 18.Which numbered term would be the fi rst to exceed 1 000 000 in this sequence?

Think WriTe

1 To fi nd the rule for the sequence, fi nd a and r and substitute them into tn = a × r n − 1.

a = 2

r = t2t1

= 62

= 3 tn = 2 × 3n − 1

2 Set up the equation to be solved. 2 × 3n − 1 = 1 000 000 3n − 1 = 500 000

3 Take logarithms to the base 10 of both sides and solve for n.

log10 (3n − 1) = log10 (500 000)(n − 1) log10 (3) = log10 (500 000)

(n − 1) = log10(500 000)

log10(3)

n − 1 = 11.9445 n = 12.9445

4 The next whole number term is the 13th. The 13th term would be the fi rst to exceed 1 000 000.

Worked example 15

A real estate agent records the number of blocks of land to be sold on a new estate each week.If the number of blocks of land sold continue to follow a geometric sequence:a prove it is a geometric sequence and thus write

the value of the common ratio, rb write a rule for the number of blocks of land,

tn, sold in week number nc calculate the number of blocks of land sold in

week number 8.

Week number Number of blocks of land sold

1 128

2 64

3 32

Think WriTe

a The fi rst term, a, is 128. To fi nd r, evaluate the ratios.

a t2t1

= 64

128

t3t2

= 32

64

= 12 =

1

2The ratios are the same, so the terms follow ageometric sequence. The ratio, r = 1

2.



b Use tn = ar n − 1 to write the rule. b tn = 128 × n

12

1

−

c 1 Use the formula in part b to fi nd the numbers of blocks of land sold in week 8 (i.e. n = 8).

c t8 = 128 × 12

7

= 1

2 Write your answer. The number of blocks of land sold in the 8th week will be 1.

exercise 5e Finding the terms of a geometric sequence 1 We 12 Find the value of the term specified for the given geometric sequences.

a Find the 10th term of the geometric sequence 2, 12, 72, 432, 2592, . . .b Find the 11th term of the geometric sequence 5, 35, 245, 1715, 12 005, . . .c Find the 18th term of the geometric sequence 8, 16, 32, 64, 128, . . .d Find the 8th term of the geometric sequence 11, 22, 44, 88, 176, . . .e Find the 11th term of the geometric sequence 5, 15, 45, 135, 405, . . .f Find the 15th term of the geometric sequence 2, 8, 32, 128, 512, . . .

2 Find the value of the term specified for the given geometric sequences in decimal form.a Find the 20th term of the geometric sequence 1.1, 2.2, 4.4, 8.8, 17.6, . . .b Find the 10th term of the geometric sequence 2.3, 2.76, 3.312, 3.9744, . . .c Find the 8th term of the geometric sequence 3.1, 8.06, 20.956, 54.4856, 141.662 56, . . .

3 Find the value of the term specified for the given geometric sequences in negative form.a Find the 9th term of the geometric sequence −2, −8, −32, −128, −512, . . .b Find the 12th term of the geometric sequence −6, −18, −54, −162, −486, . . .

4 We 13 Find the value of the term specified for the specified geometric sequences.a The 2nd term of a geometric sequence is 6 and the 5th term is 162. Find the 10th term.b The 2nd term of a geometric sequence is 6 and the 5th term is 48. Find the 12th term.c The 4th term of a geometric sequence is −32 and the 7th term is −256. Find the 14th term.

5 We 14 Evaluate the following.a The fi rst three terms of a geometric sequence are 5, 12.5 and 31.25. Which term would be the fi rst

to exceed 50 000?b The fi rst three terms of a geometric sequence are 3.2, 9.6 and 28.8. Which term would be the fi rst

to exceed 1 000 000?c The fi rst three terms of a geometric sequence are 5.1, 20.4 and 81.6. Which term would be the

fi rst to exceed 100 000?

6 We 15 The number of cells of a micro-organism, after each process of cell division, can be summarised as follows.

1, 2, 4, 8, 16

If the number of cells after each division continue to follow a geometric sequence, fi nd:a a rule for the number of cells after n divisionsb the number of cells after 12 divisions.

diGiTal doCdoc-9437SkillSHEET 5.3Solving non-linear simultaneous equations

diGiTal doCdoc-9438SkillSHEET 5.4Solving indicial equations


7 A small town is renowned for spreading rumours. All of its citizens are aware in a short time of any new rumours. The spread of the rumour can be summarised in the table below. If the number of citizens who have been told the rumour each day continues to follow a geometric sequence, find:

Day number Number of citizens in the know

1 1

2 6

3 36

a a rule for the number of citizens on day nb the number of citizens told of the rumour by day 5c on which day all 4230 citizens will know of the rumour.

8 Find the 9th term of the geometric sequence −6, 9, −13.5, 20.25, −30.375, . . . 9 Find the 8th term of the geometric sequence −6.2, 9.3, −13.95, 20.925, −31.3875, . . . 10 Find the 10th term of the geometric sequence 1

4, 1

8, 1

16, 1

32, 1

64, . . .

11 Find the 11th term of the geometric sequence 13, 2

3, 11

3, 22

3, 51

3, . . .

12 The 4th term of a geometric sequence is −81 and the 7th term is 2187. Find the 12th term.

13 The 4th term of a geometric sequence is 0.875 and the 7th term is 0.109 375. Find the 10th term.

14 The 3rd term of a geometric sequence is 1 and the 6th term is 827

. Find the 9th term.

15 The takings at a new cinema each month are recorded. If the takings each month continue to follow a geometric sequence, find:a a rule for the takings in month nb the takings in month 9.

Month number Takings

1 $10 000

2 $ 8 500

3 $ 7 225

16 A tomato grower has recorded the average height of his tomato bushes. If the height of the tomato bushes each year continues to follow a geometric sequence, find:a a rule for the height of the bushes in year nb the height of the tomato bushes in year 5 (correct to

2 decimal places)c the year in which the height of the tomato bushes will

exceed 2 m.

Year Height (m)

1 1.2

2 1.26

3 1.323

17 mC The 12th term of the geometric sequence 21, 63, 189, 567, . . . is:a 6804 B 413 343 C 1 240 029d 3 720 087 e 5 931 980 229

18 mC The 10th term of the geometric sequence 5, −10, 20, −40, 80, . . . is:a −2560 B −1280 C 1280d 5120 e 3 906 250

Chapter 5 • Arithmetic and geometric sequences 191

19 MC The 3rd term of a geometric sequence is −75 and the 6th term is −9375. The 9th term is:a −5 859 375 b −1 171 875 C −32 805D 32 805 e 234 375

20 MC The first three terms of a geometric sequence are 5.5, 7.7 and 10.78. The first term to exceed 100 would be the:a 8th b 9th C 10thD 11th e 12th

5F the sum of a given number of terms of a geometric sequenceWhen the terms of a geometric sequence are added, a geometric series is formed. So 3, 6, 12, 24, 48, . . . is a geometric sequence, whereas 3 + 6 + 12 + 24 + 48 + . . . is a geometric series.

The sum of n terms of a geometric sequence is given by Sn.Consider the general geometric sequence a, ar, ar 2, ar 3, . . . ar n − 1.

Now, Sn = a + ar + ar 2 + ar 3 + . . . + ar n − 1

Also, multiplying each term by r, rSn = ar + ar 2 + ar 3 + ar 4 + . . . + ar n

So, rSn − Sn = −a + ar n since all the other terms cancel out.So, Sn (r − 1) = a(r n − 1)

Sn = a(r n − 1)

r − 1This formula is useful if r < −1 or r > 1, for example, if r is 2, 10, 3.3, −4, −1.2.By calculating Sn − rSn instead of rSn − Sn, as we did earlier, we obtain an alternative form of the

formula. That is, Sn − rSn = a − ar n

Sn(1− r) = a(1 − r n)

Sn = a(1 − r n)

1 − r

This formula is useful if r is in between −1 and 1 (shown as −1 < r < 1).

The sum of n terms, Sn, of a geometric sequence may be calculated using

Sn = a(r n − 1)

r − 1 if r < −1 or r > 1

or

Sn = a(1 − r n)

1 − r if −1 < r < 1.

WorkeD exaMple 16

Find the sum of the first 9 terms of the sequence 0.25, 0.5, 1, 2, 4, . . .

think Write

1 Find the value of a. a = 0.25

2 Find the value of r by testing ratios of the given terms.

t2t1

= 0.5

0.25 t3

t2 =

1

0.5

= 2 = 2

t4t3

= 2

1 t5

t4 =

4

2

= 2 = 2

∴ r = 2


3 Since r > 1, use Sn = a(r n − 1)

r − 1. S9 =

0.25(29 − 1)

1

= 127.75

4 Write the answer. The sum of the fi rst 9 terms is 127.75.

Worked example 17

The 3rd term of a geometric sequence is 11.25 and the 6th term is 303.75.Find the sum of the fi rst 10 terms of the sequence correct to 1 decimal place.

Think WriTe

1 We need to fi nd a and r. We know that t3 = 11.25 and that tn = ar n − 1.

t3 = ar 2

∴ ar2 = 11.25 [1]

2 We know that t6 = 303.75 and that tn = ar n − 1. t6 = ar 5 ∴ ar5 = 303.75 [2]

3 Solve these equations simultaneously by eliminating a to fi nd r.

Equation [2] divided by equation [1]:ar5

ar2 =

303.75

11.25 r3 = 27 r = 3

4 Substitute r value into one of the equations to fi nd a, the fi rst term.

Substituting r = 3 into equation [1]: ar2 = 11.25a × 32 = 11.25

a = 11.25

9 = 1.25

5 Since r > 1, use Sn = a(r n − 1)

r − 1. S10 =

1.25(310 − 1)

2 = 36 905

6 Write your answer. The sum of the fi rst ten terms of the geometric series is 36 905.

Worked example 18

How many terms of the geometric sequence 100, 95, 90.25, 85.7375, . . . are required for the sum to be greater than 1000?

Think WriTe

1 Find a. a = 100

2 Find r. r =

95

100 = 0.95

3 Use Sn = a(1 − r n)

1 − r since r < 1. Sn =

100(1 − 0.95n)

0.05

4 Investigate how many terms are required to sum to 1000 or Sn = 1000.

100 = 100(1 − 0.95n)

0.05 = 2000(1 − 0.95n) 0.5 = 1 − 0.95n

0.95n = 0.5



5 Express 0.95 and 0.5 in terms of logarithms with a base of 10. Of course, other bases could be used. 131

2 terms are required to

sum to 1000, so we require 14 to exceed 1000, as the number of terms is discrete.

log10 (0.95n) = log10 (0.5)n log10 (0.95) = log10 (0.5)

n = log10(0.5)

log10(0.95)

= 13.513So, we require that n = 14.

exercise 5F The sum of a given number of terms of a geometric sequence 1 a We 16 Find the sum of the first 12 terms of the geometric sequence 2, 6, 18, 54, 162, . . .

b Find the sum of the fi rst 7 terms of the geometric sequence 5, 35, 245, 1715, 12 005, . . .c Find the sum of the fi rst 11 terms of the geometric sequence 3.1, 9.3, 27.9, 83.7, 251.1, . . .d Find the sum of the fi rst 12 terms of the geometric sequence −0.1, −0.4, −1.6, −6.4, −25.6, . . .

2 a We 17 The 2nd term of a geometric sequence is 10 and the 5th is 80. Find the sum of the first 12 terms of the sequence.

b The 2nd term of a geometric sequence is 15 and the 5th is 405. Find the sum of the fi rst 11 terms of the sequence.

c The 2nd term of a geometric sequence is −12 and the 5th is −768. Find the sum of the fi rst 9 terms of the sequence.

d The 3rd term of a geometric sequence is 500 and the 6th is −500 000. Find the sum of the fi rst 10 terms of the sequence.

3 a We 18 How many terms of the geometric sequence 3, 6, 12, 24, 48, . . . are required for the sum to be greater than 3000?

b How many terms of the geometric sequence 5, 20, 80, 320, 1280, . . . are required for the sum to be greater than 100 000?

c How many terms of the geometric sequence 1.2, 2.4, 4.8, 9.6, 19.2, . . . are required for the sum to be greater than 10 000?

d How many terms of the geometric sequence 120, 96, 76.8, 61.44, 49.152, . . . are required for the sum to be greater than 540?

4 mC The sum of the first 10 terms of the geometric sequence 2.25, 4.5, 9, 18, 36, . . . is closest to:a 1149.75 B 2301.75C 5318.81 d 6648.51e 8342.65

5 mC The 2nd term of a geometric sequence is −20 and the 5th is −1280. The sum of the first 12 terms of the sequence is:a −27 962 025 B −1 062 880C −15 360 d 1 062 880e 16 777 215

6 Find the sum of the first 13 terms of the geometric sequence 80, 72, 64.8, 58.32, 52.488, . . .

7 Find the sum of the first 8 terms of the geometric sequence 250, −150, 90, −54, 32.4, . . .

8 Find, correct to 1 decimal place, the sum of the first 12 terms of the geometric sequence −192, 48, −12, 3, −0.75, . . .

9 The 3rd term of a geometric sequence is 2 and the 6th is 0.016. Find, correct to 1 decimal place, the sum of the first 13 terms of the sequence.

10 The 3rd term of a geometric sequence is 90 and the 6th is 0.09. Find, correct to 1 decimal place, the sum of the first 14 terms of the sequence.

11 How many terms of the geometric sequence 600, 180, 54, 16.2, 4.86, . . . are required for the sum to be greater than 855?

diGiTal doCdoc-9439WorkSHEET 5.2


5G applications of geometric sequencesGrowth and decay of discrete variables is constantly found in real-life situations. Some examples are increasing or decreasing populations and increase or decrease in fi nancial investments. Some of these geometric models are presented here.

Worked example 19

A city produced 100 tonnes of rubbish in the year 2004. Forecasts suggest that this may increase by 2% each year. If these forecasts are true:a what will be the city’s rubbish output in 2008?b in which year will the amount of rubbish reach 120 tonnes?c what was the total amount of rubbish produced by the city in the years 2004, 2005 and 2006?This is an example of a geometric sequence where a = 100 and r = 1.02. Note that r ≠ 0.02. If this was the case, then multiplying 100 by 0.02 would result in a lesser amount of rubbish in the second year and so on. We are told that the amount of rubbish increases by 2%. That is the original amount plus an extra 2%, or: original amount + 2% of original amount = original amount × (1 + 2%) = original amount × (1 + 0.02) = 1.02 × original amount.

Think WriTe

a 1 Find the fi rst term, a. a a = 100

2 Determine the common ratio, r by expressing the new percentage as a decimal.

Increase by 2%: 100% + 2%= 102%

r = 1.02

3 Determine which term is represented by the amount of rubbish for the year 2008.

Year 2004 is the fi rst term, so n = 1.Year 2005 is the second term, so n = 2.Year 2008 is the fi fth term, so n = 5.

4 Use tn = ar n − 1 to fi nd the amount of rubbish collected in the fi fth year.

t5 = 100 × 1.025 − 1

= 100 × 1.0824 = 108.24

5 Write your response. The amount of rubbish produced in the fi fth year, or 2008, will be 108.24 tonnes.

b 1 Use tn = ar n − 1 and tn = 120. b 100(1.02)n − 1 = 120 (1.02)n − 1 = 1.2

2 Express 1.02 and 1.2 in terms of logarithms with base 10.

log10 (1.02)n − 1 = log10 (1.2)(n − 1) log10 (1.02) = log10 (1.2)

n − 1 = log10(1.2)

log10(1.02)

n − 1 = 9.207n = 10.207

3 Write your answer. During the 11th year (that is, during 2014) the amount of rubbish produced will have exceeded 120 tonnes.

c 1 We need to fi nd the sum of the fi rst 3 terms.

Use Sn = a(r n − 1)

r − 1 where n = 3.

c S3 = 100(1.023 − 1)

1.02 − 1 = 306.04

2 Write your answer. The total output of rubbish for the years 2004, 2005 and 2006 was 306.04 tonnes.



Worked example 20

A computer decreases in value each year by 15% of the previous year’s value. Find an expression for the value of the computer, Vn, after n years. Its initial purchase price is given as V1 = $12 000.

Think WriTe

1 This is a geometric sequence since there is a 15% decrease on the previous year’s value. Find a and r.

a = 12 000Decrease by 15%: 100% − 15%

= 85%r = 0.85

2 We want an expression for the value after n years. Use tn = ar n − 1 which gives the value of the nth term. (Replace tn with Vn as required).

Vn = 12 000 × (0.85)n − 1

3 Write your answer. The value of the computer is given by the expression, Vn = 12 000(0.85)n − 1.

Compound interestConsider the case where a bank pays compound interest of 5% per annum on an amount of $20 000. The amount is invested for 4 years and interest is calculated yearly.

Compound interest is named so because the interest which is earned is paid back into the account so that the next time interest is calculated, it is calculated on an increased (i.e. compounded) amount. There is a compounding effect on the money in the account.

If we calculated the amount in the account each year, we would have the following amounts.

Start $20 000After 1 year $20 000 × 1.05 = $21 000After 2 years $20 000 × 1.05 × 1.05 = $22 050After 3 years $20 000 × 1.05 × 1.05 × 1.05 = $23 152.50After 4 years $20 000 × 1.05 × 1.05 × 1.05 × 1.05 = $24 310.13

The amounts 20 000, 21 000, 22 050, 23 152.50, 24 310.13, . . . form a geometric sequence where a = 20 000 and r = 1.05.

We need to be a little careful, however, in using the formula tn = ar n − 1 in calculating compound interest. This is because the original amount in the account, that is, $20 000, in terms of the geometric sequence would be referred to as t1 or a. In banking terms, t1 would represent the amount in the account after the fi rst lot of interest has been calculated and added in.

To be clear and to avoid errors, it is best to use the following formula for compound interest.

A = PR n

where R = 1 + r

100 A = amount in the account, $ P = principal, $ r = interest rate per compounding period (e.g. per year, per quarter), % n = the number of compounding periods during the investment.

Note: Students who are studying Module 4: Business related mathematics will use this formula for compound interest.

Worked example 21

Helen inherits $60 000 and invests it for 3 years in an account which pays compound interest of 8% per annum compounding each 6 months.a What will be the amount in Helen’s account at the end of 3 years?b How much will Helen receive in interest over the 3-year period?


Think WriTe

a 1 This is an example of compound interest.

Use A = PRn, where R = 1 + r

100.

Interest is calculated each 6 months, so over 3 years, there are 6 compounding periods: n = 6. Interest is 8% per year or 4% per 6 months.So, r = 4%.

a P = 60 000 n = 6 half-years r = 4% per half-year

So, R = 1 + 4

100 = 1.04 A = PRn

= 60 000(1.04)6

= 75 919.14

2 Write your answer. At the end of 3 years, Helen will have a total amount of $75 919.14.

b 1 Interest equals the amount in the account at the end of 3 years, less the amount in the account at the start of the investment.

b Interest = Total amount − Principal = $75 919.14 − $60 000 = $15 919.14

2 Write your answer. Amount of interest earned over 3 years is $15 919.14.

Worked example 22

Jim invests $16 000 in a bank account which earns compound interest at the rate of 12% per annum compounding every quarter.

At the end of the investment, there is $25 616.52 in the account. For how many years did Jim have his money invested?

Think WriTe

1 We know the value of A, P, r and R.We need to fi nd n using the compound interest formula.Note: There are 4 quarters per annum.

A = 25 616.52P = 16 000

r = 124

= 3% per quarter

and so R = 1 + 3100

= 1.03

2 Now substitute into A = PRn. A = PRn

25 616.52 = 16 000(1.03)n

1.601 = 1.03n

3 Take logarithms with base 10 of both sides of the equation and solve for n.

log10 (1.601) = log10 (1.03n)So, log10 (1.601) = n × log10 (1.03)

n = log10(1.601)

log10(1.03)

= 0.2044

0.0128

4 Round up the number of periods to 16 to ensure the amount is reached.

n = 15.92 ≈ 16

5 Write your answer. It will take 16 compounding periods where a period is 3 months. So, it will take 48 months or 4 years.


exercise 5G applications of geometric sequences 1 We 19 A farmer harvests 4 tonnes of lucerne in his first year of production. In his business plan, he has

estimated an annual increase of 6% on his lucerne harvest.a According to this plan, how many tonnes of lucerne should he harvest in his 7th year of

production?b In which year will his harvest reach 10 tonnes?c How much will he expect to harvest in the fi rst three years?d How much in total will he harvest from the 7th year to the 10th year?

2 A taxi driver estimates that the cost of keeping her taxi on the road increases by 4.5% each year. If the cost of keeping her taxi on the road in her first year of owning a taxi was $1800:a what was the cost in the 5th year?b during which year did costs exceed $2500?c what were the total costs of keeping her taxi on the road in

the fi rst 3 years?

3 We20 The population of a town is decreasing by 10% each year. Find an expression for the population of the town, which will be referred to as Pn. The population in the first year, P1, was 10 000.

4 We21 $13 000 is invested in an account which earns compound interest at 8% p.a. compounded quarterly (i.e. calculated four times each year).a After 5 years, how much is in the account?b How much interest was earned in that period?

5 The population of the newly established town of Alansford in its first year was 6000. It is predicted that the town’s population will increase by 10% each year. If this were to be the case, find:a the population of the town in its 10th yearb in which year the population of Alansford would reach 25 000.

6 The promoters of ‘Fleago’ flea powder assert that continued application of the powder will reduce the number of fleas on a dog by 15% each week. At the end of week 1, Fido the dog has 200 fleas left on him and his owner continues to apply the powder.a How many fl eas would Fido be expected to have on him at the end of the 4th week?b How many weeks would Fido have to wait before the number of fl eas on him had dropped to less

than 50?

7 Young saplings should increase in height by 9% each year under optimum conditions. If a batch of saplings which have been planted out measure 2.2 metres in their first year:a how high should they be in their 4th year?b in which year should they exceed 5 metres in height?

8 mC A colony of ants is studied and the population of the colony in week 1 of the study is 800. If the population of the colony is expected to increase at the rate of 2% each week, then the week in which the number of ants would exceed 1000 would be closest to:a 6 B 10 C 13d 26 e 32

9 A company exported $300 000 worth of manufactured goods in its first year of production. According to the business plan of the company, this amount should increase each year by 7.5%.a How much would the company be expected to export in its

5th year?b In which year would exports exceed $500 000?c What is the total amount exported by the company in its fi rst

7 years of operation?

diGiTal doCSdoc-9440SkillSHEET 5.5relating the common ratio of a geometric sequence to percentage increase or decreasedoc-9441SpreadsheetSequences and series


10 $10 000 is invested in an account which earns compound interest at 10% per annum. Find the amount in the account after 5 years if the interest is compounded:a yearly b every 6 months c quarterly d monthly.

11 $20 000 is invested in an account earning compound interest of 10% per annum compounding quarterly. What is the amount in the account after:a 1 year? b 3 years? c 5 years? d 10 years?

12 We22 In an account earning compound interest of 8% per annum compounding quarterly, an amount of $6000 is invested. When the account is closed, there is $7609.45 in the account. For how many years was the account open?

13 Sue earns 12% interest per annum compounding quarterly on her investment of $40 000. For how many years would this investment need to operate for the amount to rise to $50 670.80?

14 An amount of $14 500 is invested in an account attracting compound interest of 6% per annum compounding quarterly. After a certain time the interest earned in the account is $1834.14. Find out for how long the amount had been invested.

5h Finding the sum of an infinite geometric sequenceIf you are 2 metres away from a wall and you move 1 metre (or halfway) towards the wall and then move 1

2 metre (or halfway again) towards the wall and continue to do this, will you reach the wall? When

will you reach the wall?

1 m

1 m

m–21 –

41 –

81

2 m

Consider the following geometric sequence:

1, 12, 1

4, 1

8, 1

16, . . . This is an infi nite geometric sequence since it continues on with an infi nite number of terms.

Each term in the sequence is half the size of the previous term, that is, r = 0.5.If we were to add n terms of this sequence together, we would have:

Sn = 1 × (1 − 0.5 n)

1 − 0.5

= 1 − 0.5 n

0.5

= 1

0.5 −

0.5 n

0.5 = 2 − 0.5n − 1

Consider 0.5n − 1 in the equation above. As n becomes very large, the term 0.5n − 1 becomes very small. Try this with your calculator.Let n = 5, 0.5n − 1 = 0.54 = 0.0625; therefore, S5 = 2 − 0.0625 = 1.9375Let n = 10, 0.5n − 1 = 0.59 = 0.001 95; therefore, S10 = 2 − 0.001 95 = 1.998 05Let n = 20, 0.5n − 1 = 0.519 = 0.000 001 9; therefore, S20 = 2 − 0.000 001 9 = 1.999 998 1

We can see that as n becomes larger, 0.5n − 1 becomes smaller. If n were to approach infi nity (note that you can never reach infi nity, you can only approach it), then the value of 0.5n − 1 would approach zero. So, Sn = 2 − 0.5n − 1 would become S∞ = 2.

It is possible to generalise this in order to fi nd the sum of an infi nite geometric sequence. We use the symbol S∞ which is referred to as the sum to infi nity of a geometric sequence.

The sum to infi nity of a geometric sequence for which −1 < r < 1 is given by:

S∞ = a

1 − r.

Concept summary


Units: 3 & 4

AOS: 1

Topic: 1

Concept: 4


Worked example 23

Find the sum to infi nity of the geometric sequence 2, 0.4, 0.08, 0.016, 0.0032, . . .

Think WriTe

1 Find a and r. a = 2

r = t2t1

= 0.4

2 = 0.2

2 As r = 0.2 satisfi es the condition−1 < r < 1, use the formula S∞ =

a

1 − r.

S∞ = a

1 − r

= 2

1 − 0.2

= 2

0.8

= 2.5

3 Write your answer. The sum to infi nity of the given sequence is 2.5.

Worked example 24

The sum to infi nity of a geometric sequence is 15 and the value of a is 10. Write the fi rst 4 terms of the sequence.

Think WriTe

1 Use the formula S∞ = a

1 − r to fi nd the value

of r. Transpose the equation to make r the

subject.

S∞ = a

1 − r

15 = 10

1 − r

1 − r = 1015

r = 1 − 23

r = 13

2 a = 10 and r = 13. Use these to generate the

terms in the sequence.The fi rst 4 terms of the sequence are

10, 103

, 109

, 1027

or 10, 313, 11

9, 10

27.

Worked example 25

The sum to infi nity of the geometric sequence is 6.25 and the value of r is 0.2. Write the fi rst 4 terms of the sequence.

Think WriTe


1 − r to fi nd the value of a. S∞ =

a

1 − r

625 = a

1 − 0.26.25 × 0.8 = a

a = 5

2 a = 5 and r = 0.2. Use these to generate the terms. The fi rst 4 terms of the sequence are 5, 1, 0.2, 0.04.


Converting recurring decimals to fractionsWe can use the sum to infinity formula to convert recurring decimals to fractions.

Worked example 26

Express .

1. 2 as a fraction.

Think WriTe

1 We need to express .

1.2 as the sum of a geometric sequence.

.1.2 = 1.222 222 222 222 2 . . . = 1 + 0.2 + 0.02 + 0.002 + 0.0002 + . . . = 1 + (0.2 + 0.02 + 0.002 + 0.0002 + . . .)

2 The terms in the bracket form an infinite geometric sequence where a = 0.2 and r = 0.1. Use the formula.

S∞ = a

1 − r.

Multiply both the numerator and the

denominator of 0.2

0.9 by 10 to eliminate the

decimal.

a = 0.2

r = 0.02

0.2 = 0.1

S∞ = a

1 − r.

0.20.2

1 0.1=

−0.2

0.9=

2

9=

3 State the final answer. So, .

1.2 = 1 + 29

= 129

Worked example 27

Express 0.645 . . . as a fraction.

Think WriTe

1 We need to express 0.645 as the sum of a geometric sequence.

0.645 = 0.645 454 5 . . . = 0.6 + 0.045 + 0.000 45 + 0.000 004 5 + . . . = 0.6 + (0.045 + 0.000 45 + 0.000 004 5 + . . .)

2 The terms in the bracket form an infinite geometric sequence where a = 0.045 and r = 0.01.

a = 0.045

r = 0.000 45

0.045 = 0.01


1 − r to express 0.045

as a fraction first.

Multiply both the numerator and the

denominator of 0.045

0.99 by 1000 to eliminate

the decimal and simplify.

S∞ = a

1 − r

0.0450.045

1 0.01=

−

= 0.045

0.99

= 45

990

= 1

22


4 0.645 0.6 0.045.= +Write both components as fractions and simplify.

0.645 0.6 0.045= += 6

10 + 1

22

= 66

110 +

5

110

5 Write your answer. So, 0.645 71110

=

Worked example 28

An injured rabbit attempts to crawl back to its burrow. It moves 30 metres in the fi rst hour, 21 metres in the second hour and 14.7 metres in the third hour and so on. If the burrow is 200 metres away, will the rabbit make it back?

Think WriTe

1 Determine what sort of sequence we have.

t1 = 30, t2 = 21 and t3 = 14.7

Now 2130

= 0.7

and 14.721

= 0.7

So, we have a geometricsequence where a = 30, r = 0.7.

2 Find the value of S∞ = a

1 − r. S∞ =

30

1 − 0.7 = 100

3 Write your answer. The rabbit will cover a total of 100 metres. Since the rabbit hole is 200 metres away, the rabbit won’t make it.

exercise 5h Finding the sum of an infinite geometric sequence 1 We23 Find the sum to infinity of the following geometric sequences.

a 50, 25, 12.5, 6.25, 3.125, . . . b 20, 16, 12.8, 10.24, 9.192, . . .c 1, 1

3, 1

9, 1

27, 1

81, . . . d 1, 1

5, 1

25, 1

125, 1

625, . . .

e 3, −0.6, 0.12, −0.024, 0.0048, . . . f −50, 5, −0.5, 0.5, −0.05, . . . 2 We24 Write the first 3 terms of the geometric sequence for which:

a r = 0.6 and S∞ = 25 b r = 0.25 and S∞ = 8c r = 0.9 and S∞ = 120 d r = −0.2 and S∞ = 31

3

e r = −0.8 and S∞ = 5 f r = 0.2 and S∞ = −7.5.

3 We25 Write the first 3 terms of the geometric sequence for which:

a a = 12.5 and S∞ = 25 b a = 12.5 and S∞ = 50

c a = 48 and S∞ = 120 d a = 249 and S∞ = 32

3.

4 We26, 27 Express each of the following recurring decimals as fractions.

a 0.5 b .

0.4 c .

1.3 d .

3.7

e 8.666 666 666 . . . f . .

0.14 g 0.529 h 1.321.

5 We28 A defiant child walks 10 metres towards his mother in the first minute, 4 metres in the second minute and 1.6 metres in the third minute. If the child continues to approach in this same pattern, and if his mother is standing stationary, 20 metres from the child’s initial position, will the child ever reach the mother?


6 A failing machine produces 35 metres of spouting in the first hour, 21 metres in the second hour and 12.6 in the third hour. If this pattern continues and 280 metres of spouting is required, how far short of the quota will the machine fall?

7 A nail penetrates 20 mm with the first hit of a hammer, 12 mm with the 2nd hit and 7.2 mm with the 3rd. If this pattern continues, will the 50 mm long nail ever be completely hammered in?

8 A woman establishes a committee to raise money for a hospital. It raises $40 000 in the first year, $36 800 in the 2nd year and $33 856 in the third year. If the fundraising continues in this pattern, how far short will they fall in raising $1 000 000?

9 A will of a recently deceased woman specifies how her money is to be donated to a charity. Her total wealth of $12.5 million is to be donated for eternity with the first donation of $1 million in the first year.a What fraction of this fi rst donation should be donated for the second year and subsequent years?b Write the value of the donations for each of the fi rst 5 years.c How much will be donated after 10 years?

5i Contrasting arithmetic and geometric sequences through graphsWhen discrete variables are presented graphically some distinct features may be evident. This is especially so for discrete variables that have an arithmetic or geometric pattern.

arithmetic patternsArithmetic patterns are distinguished by a straight line or a constant increase or decrease.

Val

ue o

f te

rm t n

Term n54321

d is positive

Val

ue o

f te

rm t n

Term n54321

d is negative

An increasing pattern or a positive common difference gives an upward straight line.

A decreasing pattern or a negative common difference gives a downward straight line.

Geometric patternsGeometric patterns are distinguished by a curved line or a saw form.

Val

ue o

f te

rm t n

Term n54321

Val

ue o

f te

rm t n

Term n54321

An increasing pattern or a positive common ratio greater than 1 (r > 1) gives an upward curved line.

A decreasing pattern or a positive fractional common ratio (0 < r < 1) gives a downward curved line.

inTeraCTiViTYint-0186

Contrasting arithmetic and geometric

sequences


Val

ue o

f te

rm t n

Term n

54321

+

−

Val

ue o

f te

rm t n

Term n

54321

+

−An increasing saw pattern occurs when the common ratio is a negative value less than −1 (r < −1).

A decreasing saw pattern occurs when the common ratio is a negative proper fraction (−1 < r < 0).

Worked example 29

On the graph at right, the fi rst 5 terms of a sequence are plotted. State whether the sequence could be arithmetic or geometric and give the value of a and the value of either d or r.

Think WriTe

1 Examine the difference between the value of each of the terms. In each case, they are the same, that is, 10.

There is a constant difference between each successive term so the straight line graph shows an arithmetic sequence.

2 Find a. t1 = 40, so a = 40.

3 Find d. t2 = 30 and t1 = 40.d = t2 − t1 = −10.

Worked example 30

Calculate the total amount in an account if $10 000 is invested for 5 years and earns:a simple interest of 10% per annumb compound interest of 10% per annum compounding yearly.

c For each of the above cases, graph, on the same set of axes, the amount in the account over the fi ve years. Use your graph or calculations to calculate the difference between the accounts after 4 years.

Think WriTe

a Calculate how much is in the account earning simple interest at the end of each of the fi ve years.

a After 1 year, amount in account = 10 000 + 10% of 10 000 = 10 000 + 1000 = $11 000After 2 years, amount in account = 10 000 + 2 × 10% of 10 000 = $12 000After 3 years = $13 000After 4 years = $14 000After 5 years = $15 000

Val

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rm

Term number54321

1020304050

0


b Calculate the amount in the account earning compound interest at the end of each of the 5 years using A = PRn where R = 1 +

r

100(that is, 1 + 10

100 = 1.1).

b After 1 year, amount in account = 10 000 × 1.11

= $11 000After 2 years, amount in account = 10 000 × 1.12

= $12 100After 3 years = $13 310After 4 years = $14 641After 5 years = $16 105

c 1 Draw the graphs of the amount in the account earning simple interest (straight line) and the amount in the account earning compound interest (curved line) on the same set of axes.

c

Number of years invested (n)3 4 5210

16 00015 000

13 00012 00011 000

10 000

14 000

17 000

Am

ount

in a

ccou

nt (

$) Compound interest

Simple interest

2 Use the values calculated for the end of the fourth year.

Difference in amounts = 14 641 − 14 000 = $641

3 Write your answer. After 4 years, the compound interest account earns an extra $641 in interest.

exercise 5i Contrasting arithmetic and geometric sequences through graphs 1 We29 On the graph below, the first five terms of a sequence are plotted.

Val

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f te

rm

Term number543210

12345

State whether the sequence could be arithmetic or geometric and give the value of a and the value of either d or r.

2 On the graph below, the first five terms of a sequence are plotted.

Val

ue o

f te

rm

Term number54321

2468

1012

0



3 On the graph below, the first five terms of a sequence are plotted.

Val

ue o

f te

rm

Term number54321

102030405060

0


4 On the graph at right, the first five terms of a sequence are plotted.State whether the sequence could be arithmetic or geometric

and give the value of a and the value of either d or r.

5 Draw a graph showing the first 8 terms of each of the following sequences:

a arithmetic, a = 7, d = 2 b geometric, a = 5, r = 12

c arithmetic, a = −14, d = 3.5 d arithmetic, a = 32, d = −5e geometric, a = 12, r = 10

3.

6 mC On the graph at right, the first five terms of a sequence are plotted. The sequence could be described by which one of the following?a Arithmetic sequence with a = 10 and d = 10B Arithmetic sequence with a = 10 and d = 0.5C Geometric sequence with a = 10 and r = 0.5d Geometric sequence with a = 10 and r = 2e Geometric sequence with a = 10 and r = 1.5

7 mC On the graph at right, the first five terms of a sequence are plotted. The sequence could be described by which one of the following?a Arithmetic sequence with a = 10 and d = 20B Arithmetic sequence with a = 10 and d = −20C Geometric sequence with a = 10 and r = 10d Geometric sequence with a = 10 and r = −20e Geometric sequence with a = 10 and r = −1

8 We30 An amount of $5000 is invested for 3 years and earns:a simple interest of 10% per annumb compound interest of 10% per annum compounding yearly.On the same set of axes, plot points showing the amount in each account at the end of each of the 3 years.

9 An amount of $100 000 is invested for 3 years and earns:a simple interest of 15% per annumb compound interest of 15% per annum compounding yearly.On the same set of axes, plot points showing the amount in each account at the end of each of the 3 years.

10 On the same set of axes, sketch the graphs of the sequences with the rule un = 10n and vn = 10 × 1.5n − 1. Use your graph to decide for how many of the first five terms un is greater than vn.

11 On the same set of axes, sketch the graphs of the sequences with the rule un = 120 − 20n and vn = 100 × 0.8n − 1. Use your graph to decide for how many of the first five terms un is greater than vn.

Val

ue o

f te

rm

Term number54321

5101520

0

Val

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f te

rm

Term number54321

306090

120150180

0

Val

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f te

rm

Term number54321

−10−505

10


SummaryRecognition of arithmetic sequences

• Anarithmeticsequenceisasequenceofnumbersforwhichthedifferencebetweensuccessivetermsisthesame.

•Givenanarithmeticsequence,identify: thefirstterm,a andthecommondifference,d=t2−t1.

• Givenanunspecifiedsequence,establishwhetheritisarithmeticbytestingalltermsforacommondifference: d=t2−t1=t3−t2=t4−t3=. . .

Finding the terms of an arithmetic sequence

• tn=a+(n−1)dwhere tnisthenthterm. aisthefirstterm disthecommondifference.

The sum of a given number of terms of an arithmetic sequence

•Aseriesisthesum of termsinasequence.•Snisthesumofthefirstntermsinseries,forexample,S25representsthesumofthefirst25terms.•Givenanumberoftermsinaseries,n,firstterm,aandthelastterm,l,use:

Sn=n

2(a+l).

•Givenanumberoftermsinaseries,n,firstterm,aandthecommondifference,d,use:

Sn=n

2[2a+(n−1)d ].

Recognition of geometric sequences

•Ageometricsequenceisasequenceofnumbersforwhichtheratioofsuccessivetermsisthesame.•Givenageometricsequence,identify: thefirstterm,a andthecommonratio,r=t2

t1.

• Givenanunspecifiedsequence,establishwhetheritisgeometricbytestingalltermsforacommon

ratio,r=t2t1=t3

t2=t4

t3=...

Finding the terms of a geometric sequence

• tn=ar n −1

where tnisthenthterm aisthefirstterm risthecommonratio.

The sum of a given number of terms of a geometric sequence

• 2,4,8,6,32,64isafinitegeometricsequence.• 2+4+8+16+32+64isafinitegeometricseries.• Thesumofnterms,Sn,ofageometricsequencemaybecalculatedusing:

Sn=a(r n−1)

r−1ifr>1orr<−1(forexample,r= −2or 3

2

−

or+2or+4.5).

or

Sn=a(1 −r n)

1 −rif−1<r<1(forexample,r=0.2or1

8or−0.25).

Geometric growth •Growthorincreaseisexpressedasapercentageincrease.

• Commonratio,r=1+%increase

100.

• rvaluesaregreaterthan1,forexample,an8%increasegivesr=1.08.

Geometric decay •Decayordecreaseisexpressedasapercentagedecrease.

• Commonratio,r=1−%decrease

100.

• rvaluesarelessthan1,forexample,an8%decreasegivesr=0.92.


Compound interest • A = PRn where R = 1 + r

100and A = amount in the account, $

P = principal, $r = interest rate per compounding period (e.g. per year, per quarter), %n = the number of compounding periods during the investment.

Finding the sum of an infinite geometric sequence

• For decreasing or decaying geometric series, the sum of an infi nite number of terms approaches a fi nite sum.

• The sum to infi nity of a geometric sequence for which −1 < r < 1 is given by:

S∞ = a

1 − r.

Contrasting arithmetic and geometric sequences through graphs

• Arithmetic patterns are distinguished by a straight line.

Val

ue o

f te

rm t n

Term n54321

d is positiveV

alue

of

term

t n

Term n54321

d is negative

An increasing pattern or a positive common difference gives an upward straight line.

A decreasing pattern or a negative common difference gives a downward straight line.

• Geometric patterns are distinguished by a curved line or a saw form.

Val

ue o

f te

rm t n

Term n54321

Val

ue o

f te

rm t n

Term n54321

An increasing pattern or a positive common ratio greater than 1 (r > 1) gives an upward curved line.

A decreasing pattern or a positive fractional common ratio (0 < r < 1) gives a downward curved line.

Val

ue o

f te

rm t n

Term n

54321

−

+

Val

ue o

f te

rm t n

Term n

54321

+

−An increasing saw pattern occurs when the common ratio is a negative value less than −1 (r < −1).

A decreasing saw pattern occurs when the common ratio is a negative proper fraction (−1 < r < 0).


Chapter reviewmUlTipleChoiCe

1 Which of the following could be the first 5 terms of an arithmetic sequence?a 1, 2, 4, 8, 12, . . . B 3, 3, 6, 6, 9, . . . C 56, 57, 58, 59, 60, . . .d −5, 5, 10, 15, 20, . . . e 1, 4, 9, 16, 25, . . .

2 For the sequence −3.6, −2.1, −0.6, 0.9, 2.4, . . ., it is true to say it is:a an infi nite sequence with a = −3.6 and d = −0.15B an infi nite sequence with a = −3.6 and d = 1.5C an infi nite sequence with a = −0.15 and d = −3.6d a fi nite sequence with a = −0.15 and d = −3.6e a fi nite sequence with a = −3.6 and d = 0.15.

3 For the arithmetic sequence, −1, 1, 3, 5, 7, . . . the value of a, the value of d and the rule for the sequence are given respectively by:a a = −1, d = 2, tn = −3 + 2n B a = −1, d = 2, tn = −3 − n C a = 1, d = −1, tn = 2 − nd a = 2, d = −1, tn = 3 − n e a = 2, d = −1, tn = 3 − n

4 The 43rd term of the arithmetic sequence −7, 2, 11, 20, 29, . . . is:a −327 B −243 C 371 d 380 e 387

5 The 3rd term of an arithmetic sequence is 3.1 and the 7th term is −1.3. The value of the 31st term is:a −153.7 B −27.7 C 28.9 d 38.3 e 157.9

6 The sum of the first 24 terms of the sequence −16, −12, −8, −4, 0, . . . is:a 720 B 912 C 1344 d 1440 e 1488

7 The first term of an arithmetic sequence is 14 and the 3rd is 8. The sum of the first 30 terms of the sequence is:a −1770 B −1095 C −885 d 1725 e 2190

8 There is a geometric sequence for which a = 3 and r is a negative number. We can be certain that:a r is a fraction less than 1 B the 3rd term will be a positive numberC the 3rd term will be greater than the 1st term d only one number in the sequence is positivee the 4th term will be greater than the 3rd term

9 Which of the following is a geometric sequence?

a 2, −2, 2, −2, 2, . . . B 2, 4, 6, 8, 10, . . . C 1, 13,

−19

, 127

, −181

, . . .d 4, −4, 2, −2, 1, . . . e 100, 10, 0.1, 0.01, 0.001, . . .

10 The 19th term of the geometric sequence 3.25, 6.5, 13, 26, 52, . . . is:a 425 984 B 851 968 C 1 703 936 d 41 978 243 e 3 272 883 098

11 The 3rd term of a geometric sequence is 19.35 and the 6th is 522.45. The 12th term of the sequence is:a 16 539.15 B 417 629.75 C 126 955.35 d 380 866.05 e 1 142 598.15

12 The first 3 terms of a geometric sequence are 2.25, 4.5, 9. The first term to exceed 1000 is:a t9 B t10 C t11 d t12 e t13

13 The sum of the first 10 terms of the geometric sequence 8, 4, 2, 1, 12, . . . is closest to:

a 15 B 16 C 17 d 18 e 20

14 The 3rd term of a geometric sequence is 0.9 and the 6th is 7.2. The sum of the first 12 terms of the sequence is closest to:a −2 B 122 C 921 d 4122 e 8190

15 A tree increases in height each year by 5%. If it was 1.2 m high in its first year, in its 6th year its height would be closest to:a 1.53 m B 1.61 m C 5.5 m d 9.11 m e 3750 m

16 Profits in a company are projected to increase by 8% each year. If the profit in the first year was $60 000, in which year could a profit in excess of $100 000 be expected?a year 6 B year 7 C year 8 d year 9 e year 10

17 The sum to infinity of the geometric sequence 1, 4

5,

16

25,

64

125,

256

625, . . . is:

a 15

B 54

C 145

d 4 e 5

18 The first term of the geometric sequence for which r = −0.5 and S∞ = 513 is:

a −1 B −213

C 223

d 8 e 1023


19 The first five terms of a sequence are plotted on the graph at right. The sequence could be described by which of the following?a Arithmetic sequence with a = 50 and d = 25B Arithmetic sequence with a = 50 and d = 0.5C Geometric sequence with a = 50 and r = 0.5d Geometric sequence with a = 50 and r = 1.5e Geometric sequence with a = 50 and r = 2

20 The first five terms of a sequence are plotted on the graph at right. The sequence could be described by which of the following?a Arithmetic sequence with a = 10 and d = −5B Arithmetic sequence with a = 10 and d = −0.5C Geometric sequence with a = 10 and r = 5d Geometric sequence with a = 10 and r = −5e Geometric sequence with a = 10 and r = 0.5

ShorT anSWer

1 For the sequences below, state whether or not they are an arithmetic sequence. If they are, give the value of a and d.

a −123, −23, 77, 177, 277, . . . b −514, −21

4, 3

4, 33

4, 63

4, . . .

2 If the second term of an arithmetic sequence is −5 and the fifth term is 16, which term in the sequence is equal to 226?

3 Blood donations at a suburban location increase by 40 each year. If there are 520 donations in the first year:a how many donations are made in the 15th year?b what is the total number of donations made over those 15 years?

4 For each of the sequences below, state whether or not they are a geometric sequence. If they are, state the value of a and r.

a 5, 52, 5

4, 5

8, 5

16, . . .

b −700, −70, −7, 7, 70, . . . 5 The amount of garbage (in tonnes) collected in a particular area by the local council each year is

recorded over 3 successive years.

Year number Amount of garbage (tonnes)

1 7.2

2 8.28

3 9.522

If the amount collected each year were to continue to follow a geometric sequence:

a write a rule for the amount of garbage, tn, which would be collected in the area in year nb how much garbage would be collected in the 8th year? (Answer correct to 2 decimal places.)c in which year would the amount of garbage collected exceed 30 tonnes?

6 How many terms of the geometric sequence 164, 131.2, 104.96, 83.968, 67.1744, . . . are required for the sum to exceed 800?

Val

ue o

f te

rm

Term number54321

50100150200250300

0

Val

ue o

f te

rm

Term number54321

−10−505

1015

−15


7 Andrew invests $25 000 in an account earning compound interest of 10% per annum compounding quarterly.a Find the amount in the account after 3 years.b Find how long it would take to have $40 965.41 in his account.

8 Express 3.7 as a fraction.

9 The batteries in a toy soldier are running down. The toy soldier marches 50 cm in the first minute, 30 cm in the second minute, 18 cm in the next and so on. By how much does the toy soldier fall short of marching 1.5 m?

10 On the same set of axes, sketch the graph of the sequence with the rule:a un = 10n b vn = 10 × 2n − 1.

exTended reSponSe

Task 1 1 Consider the geometric sequence 1, 3, 9, . . . , whose common ratio is 3.

a Subtract successive terms to form the sequence 2, 6, . . . Is this a geometric sequence as well and, if so, what is its common ratio?

b Add successive terms to form the sequence 4, 12, . . . Is this a geometric sequence as well and, if so, what is its common ratio?

c Repeat parts a and b for multiplication and division.d Prove your result of part a for any geometric sequence.

2 In this problem we are comparing simple and compound interest.Consider a bank, which offers a simple interest rate of 5% per annum on an investment of $100.a What is the value of the investment for each of the fi rst 5 years?b Consider another bank, which offers compound interest at the same rate of 5%. What is the value

of the investment for each of the fi rst 5 years?c When will the value of the investment in part b be twice as much as the investment in part a?

Task 2 1 A newly established quarry produces

crushed rock for the building of roads and freeways. The amount of crushed rock (in tonnes) it produces increases by 3

12 tonnes each month and its

production for the first 3 months of operation is shown below.

MonthCrushed rock

produced (tonnes)

1 8

2 11.5

3 15

a Write the amount of crushed rock produced in the 4th month.

b Write a rule for tn, the amount of crushed rock produced in month n, expressed in terms of n, the nth month.

c Write the amount of crushed rock produced in the 60th month.d During which month will the amount of crushed rock coming from the quarry exceed

100 tonnes?e The local council has ordered that after a total of 3050 tonnes of crushed rock has been extracted

from the quarry, an environmental impact survey must be completed. After how many months will that happen?


2 The amount of crushed rock produced each month at a second quarry is shown below.

Month Crushed rock produced (tonnes)

1 10

2 11

3 12.1

Given that production at this quarry increases geometrically, fi nd:a the common ratio, rb a rule for the amount of crushed rock produced, tn, in tonnes, expressed in terms of the number of

months, nc the amount of crushed rock produced in the 5th monthd in which month the amount of crushed rock produced exceeds 30 tonnese the total amount of crushed rock produced by the quarry in its fi rst year of operation.

3 During its first month of production, the second quarry produces more crushed rock than the first quarry. In the months after that, however, the first quarry produced more crushed rock than the second quarry. After how many months does the second quarry produce more than the fi rst quarry again?

Units: 3 & 4

AOS: 1

Topic: 1

Practice VCE exam questions

Use StudyON to access all exam questions on this topic since 2002.

diGiTal doCdoc-9442Test YourselfChapter 5


ICT activitiesChapter openerdiGiTal doC

• 10 Quick Questions doc-9435: Warm up with a quick quiz on arithmetic and geometric sequences. (page 171)

5B Finding the terms of an arithmetic sequenceTUTorial

• We 5 eles-1268: Watch a worked example on finding the value of a term in an arithmetic sequence given its value and the beginning of an arithmetic sequence. (page 176)

inTeraCTiViTY• Number patterns int-0007: Recognise the relationship between two

variables by observing patterns. (page 175)

5C The sum of a given number of terms of an arithmetic sequencediGiTal doC

• WorkSHEET 5.1 doc-9436: Recognise arithmetic sequences and series. (page 182)

TUTorial• We 8 eles-1269: Watch a tutorial on finding the sum of an

arithmetic sequence using the formula. (page 180)

5e Finding the terms of a geometric sequencediGiTal doCS

• SkillSHEET 5.3 doc-9437: Practise solving non-linear simultaneous equations (page 189)

• SkillSHEET 5.4 doc-9438: Practise solving indicial equations (page 189)

TUTorial• We 14 eles-1270: Learn how to find the value of a term in

a geometric sequence using a CAS calculator and by using a spreadsheet. (page 188)

5F The sum of a given number of terms of a geometric sequencediGiTal doC

• WorkSHEET 5.2 doc-9439: Recognise geometric sequences and series. (page 193)

TUTorial• We 17 eles-1271: Watch a tutorial on finding the value of the

sum of a geometric sequence given two non-consecutive terms. (page 192)

5G applications of geometric sequencesdiGiTal doCS

• SkillSHEET 5.5 doc-9440: Practise relating the common ratio of a geometric sequence to percentage increase or decrease. (page 197)

• Spreadsheet doc-9441: Investigate graphs of arithmetic sequences and series. (page 197)

TUTorial• We 19 eles-1332: Watch a worked example on applying the

concepts involved in geometric sequences to real life. (page 194)

5i Contrasting arithmetic and geometric sequences through graphsinTeraCTiViTY

• Contrasting arithmetic and geometric sequences int-0186: Consolidate your understanding of arithmetic and geometric sequences. (page 202)

Chapter reviewdiGiTal doC

• Test Yourself doc-9442: Take the end-of-chapter test to test your progress. (page 211)

To access eBookPLUS activities, log on to www.jacplus.com.au


Answers CHAPTER 5

ariThmeTiC and GeomeTriC SeQUenCeS exercise 5a recognition of arithmetic sequences 1 a, c, i 2 a a = 2, d = 5

c a = 2, d = 2i a = 10, d = 10

3 a, c, d, e 4 a a = −123, d = 100

c a = −7, d = 6d a = −67, d = 40e a = 5, d = −7

5 a, c, f 6 a a = 0.7, d = 0.3

c a = −3.5, d = 1.5f a = −5.2, d = −0.8

7 a, b, c, d 8 a a = 1

2, d = 1 b a = 1

4, d =

1

2

c a = 15, d = 2

5d a = 3

4

−, d = 3

4

9 a, d, e 10 a a = 2, d = 2

d a = not specifi ed, d = 2e a = 8, d = 8

11 a 7th b 7th c 8th d 5th 12 5, 12, 19, 26, 33, 40, 47, 54, 61, 68 13 212, 221, 230, 239, 248 14 B 15 D

exercise 5B Finding the terms of an arithmetic sequence 1 a 122 b 38 c 2900

d 149 e −219 f 549.9 2 a 103 b 1777 c −60 d −217 3 a 24th b 28th c 34th d 279th 4 tn = 13 + 10n 5 tn = 37 + 3n 6 a tn = 5 + 10n b 10 7 A 8 B 9 D

10 −26.8 11 121

5 12 78.4

13 241

2 14 31st 15 44th

16 E 17 10.9 metres 18 101 19 tn = 3 + 4n 20 a tn = 8.5 + 3.5n

b 358.5 metres

exercise 5C The sum of a given number of terms of an arithmetic sequence 1 a 820 b 5950

c 667 d −928 2 3320 3 107.5 4 a 735 b 634.8

c 396.5 d 182.25 5 a 56 b 2, 5, 8 6 a 170 b −4, 2, 8 7 B 8 B 9 13 680 10 364 11 11 562 12 a 31 b 5.5, 7, 8.5 13 a −46 b 11, 8, 5

14 $9375 15 $414 000 16 363.8 cm 17 1135 18 a $10 600 b $136 000 19 a $155 b $2150

exercise 5d recognition of geometric sequences 1 a, b, c, f, g 2 a a = 1, r = 2 b a = 2, r = 3 c a = 1, r = 4 f a = 3, r = 2 g a = 5, r = 2 3 a, d, e, h 4 a a = −1, r = 2 d a = −2, r = 3 e a = −4, r = 2 h a = −5, r = 4 5 a, c, d 6 a a = 1, r = −2 c a = 4, r = −3 d a = 7, r = −1 7 a, c, f 8 a a = −2, r = −2 c a = −3, r = 5

f a = −6, r = −10 9 a, c, d, e 10 a a = 1.2, r = 2 c a = 2.25, r = 2 d a = 7, r = 1

2 e a = 10, r = 1.2

11 a, c, d, f

12 a a = 1

2, r = 1

2 c a = 1

4, r = 1

5

d a = 1

3, r = 2 f a = 1, r =

−1

3

13 D 14 D 15 D 16 C 17 a Various answers b a = $100, r = 1.1 18 a 1, 2, 4, 8, 16 b 2 c 2048 19 20 million, 10 million, 5 million,

21

2 million, 11

4 million, 625 000, 312 500

20 a Greater than b 0 and 1 c Equal to d −1

e −1 and 0

exercise 5e Finding the terms of a geometric sequence 1 a 20 155 392 b 1 412 376 245 c 1 048 576 d 1408 e 295 245 f 536 870 912 2 a 576 716.8 b 11.867 494 81

c 2489.861 155 3 a −131 072 b −1 062 882 4 a 39 366 b 6144 c −32 768 5 a 12th b 13th c 9th 6 a tn = 2n − 1 b 2048 7 a tn = 6n − 1 b 1296 c 6 8 −153.773 437 5 9 105.932 812 5 10

1

2048

11 3411

3

12 −531 441

13 0.013 671 875 14 64

729

15 a tn = 10 000 × 0.85n − 1 b $2724.91

16 a tn = 1.2 × 1.05n − 1

b 1.46 m c 12 17 D 18 A 19 B 20 C

exercise 5F The sum of a given number of terms of a geometric sequence 1 a 531 440 b 686 285 c 274 576.3 d −559 240.5 2 a 20 475 b 442 865 c −262 143 d −4 545 454 546 3 a 10 b 8 c 14 d 11 4 B 5 A 6 596.65 7 153.6256 8 −153.6 9 62.5 10 10 000.0 11 5

exercise 5G applications of geometric sequences 1 a 5.67 b 17th year c 12.7 tonnes d 24.82 2 a $2146.53 b Year 9 c $5646.65 3 Pn = 10 000 × 0.9n − 1

4 a $19 317.32 b $6317.32 5 a 14 147 b 16th year 6 a 123 b 10 weeks 7 a 2.85 m b Year 11 8 C 9 a $400 640.74 b Year 9 c $2 636 196.56 10 a $16 105.10 b $16 288.95 c $16 386.16 d $16 453.09 11 a $22 076.26 b $26 897.78 c $32 772.33 d $53 701.28 12 3 years 13 2 years 14 2 years

exercise 5h Finding the sum of an infinite geometric sequence 1 a 100 b 100 c 3

2

d 5

4 e 2.5 f −45 5

11

2 a 10, 6, 3.6 b 6, 1.5, 0.375 c 12, 10.8, 9.72 d 4, −0.8, 0.16 e 9, −7.2, 5.76 f −6, −1.2, −0.24 3 a 12.5, 6.25, 3.125 b 12.5, 9.375, 7.031 25

c 48, 28.8, 17.28d 24

9, 22

27, 22

81

4 a 5

9 b 4

9 c 11

3

d 37

9 e 82

3 f 14

99

g 262

495 h 1321

999

5 No — falls short by 31

3 metres

6 192.5 m 7 Yes 8 $500 000

9 a 23

25

b $1 000 000, $920 000, $846 400, $778 688, $716 392.96

c $7 070 144.32


exercise 5i Contrasting arithmetic and geometric sequences through graphs 1 Arithmetic sequence with a = 0 and d = 1 2 Arithmetic sequence with a = 10 and d = −2 3 Geometric sequence with a = 10 and r = 1.5 4 Geometric sequence with a = 20 and r = 0.5 5 a

3 4 5 6 7 821

20

15

10

5

30

25

n

tn

0

b

3 4 5 6 7 821

4

3

2

1

6

5

n

tn

0

c

3 4 5 6 7 821

−15

5

0

−5

−10

15

10

n

tn

d

3 4 5 6 7 821

20151050

−5

40

3035

25

n

tn

e

4 × 104

2 × 104

8 × 104

6 × 104

n

tn

3 4 5 6 7 8210

6 D 7 E 8

Am

ount

($)

Year321

5500

6000

6500

5000

LegendSimple interestCompound interest

0

9

Am

ount

($)

Year321

140 000

130 000

120 000

110 000

160 000

150 000

Legend

0

Simple interestCompound interest

10 3 terms 11 0 terms

ChapTer reVieW

mUlTiple ChoiCe 1 C 2 B 3 A 4 C 5 B 6 A 7 C 8 B 9 A 10 B 11 D 12 B 13 B 14 C 15 A 16 C 17 E 18 D 19 D 20 A

ShorT anSWer

1 a Yes, a = −123, d = 100 b Yes, a = −51

4, d = 3

2 Term number 35

3 a 1080 b 12 000 4 a Yes, a = 5, r = 1

2 b No

5 a tn = 7.2 × 1.15n − 1 b 19.2 tonnes c Year 12 6 17 7 a $33 622.22 b 5 years 8 37

9

9 25 cm

10

Am

ount

($)

Term number (n)321

40

30

20

10

4

60

50

80

70

Legend

Vn = 10 × 2n − 1

Un = 10n

0

exTended reSponSe

Task 1 1 a Common ratio is 3.

b Common ratio is 3.c Multiplication: common ratio is 9 (32);

division: common ratio is 1 (30).d The ratio of the second term to the fi rst

term, after the subtraction, is

ar ar

ar a

ar

a rr

( 1r( 1r )

( 1a r( 1a r ).

2 −−

=( 1−( 1

( 1−( 1=

2 a $105, $110, $115, $120, $125b $105, $110.25, $115.76, $121.55,

$127.63c 35 years

Task 2 1 a 181

2 tonnes b tn = 4.5 + 3.5n

c 214.5 tonnes d 28th month e 40 months 2 a 1.1 b tn = 10 × 1.1n − 1

c 14.641 tonnes d 13th month e 213.84 tonnes 3 24th month

Arithmetic and geometric sequences - Hawker Maths 2022

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Transcript of Arithmetic and geometric sequences - Hawker Maths 2022