CHAPTER 6 ARITHMETIC AND GEOMETRIC SERIES

_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________
CHAPTER OVERVIEW

Arithmetic series and Geometric Series

Arithmetic Geometric Applications
series series

nth term sum nth term sum Arithmetic Geometric
formula formula

finite infinite
series series

6.1 Arithmetic Series Explanation

No Terms A set of real number a1, a2 , a3...an ... which is
1 Sequence arranged (ordered )

2 Finite sequence ex: 21,22 ,23 ,24 ,...215
3 Infinite sequence
4 Series ex: 21,22 ,23 ,24 ,...215,...

Sum of a sequence
 a1  a2  a3  an finite (terminal)
series
 a1  a2  a3 infinite (unending) series

Definition of Arithmetic Sequence

A sequence T1,T2 ,T3 ,,Tn , is called an arithmetic sequence or arithmetic
progression, if there exists a constant d, called the common difference, such that
d  Tn  Tn1 .

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nth Term Formula of Arithmetic Sequence

A sequence for which any element except the first can be obtained by adding a
constant d to the preceding element.

T1  a,
T2  a  d

T3  a  d   d  a  2d
T4  a  2d   d  a  3d



Tn  a  n 1d

Sum to n terms of an arithmetic series

Let Sn be the sum of the first n terms of an arithmetic series whose last term is l.

Sn  a  a  d  a  2d  l  d  l

Reversing order Sn  l  l  d  l  2d  a  d  a

Adding, 2Sn  a  l a  l a  l a  l a  l

Since there are n terms, 2Sn  na  l

Sn  n a  l
2

Substituting, l  a  n 1d

Sn  n a  a  n 1d 
2

Therefore, Sn  n 2a  n 1d 
2

Therefore the relation between Tn and Sn is Tn  Sn  Sn1

Example 1

Which of the following can be the first four terms of arithmetic sequence? If so, write the
explicit formula for the sequence.

a) 10,8.5, 7,5.5,
b) 1, 2,3,5,

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_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________

Solution:
a) d1  8.5 10  1.5
d2  7  8.5  1.5
d3  5.5  7  1.5
Since the sequence has a common difference d  1.5, so it is an arithmetic sequence.

Therefore, the explicit formula, Tn  a  n 1d
 10  n 11.5

 10 1.5n 1.5
 11.5 1.5n

b) d1  2 1  1
d2  3 2 1
d3  5  3  2
Since there is no common difference, so it is not an arithmetic sequence.

Example 2

a. If the first and tenth term of an arithmetic sequence are 3 and 30, respectively, find the
twentieth term of the sequence.

b. Given the arithmetic sequence 1, 2 , 1 ,0,  1 ,. Find the term that equal  2 .
33 3

Solution:
a. Given a  3 and T10  30 .

By using Tn  a  n 1d
30  3  10 1d

d 3

Therefore, T20  3  20 13

 60

b. a  1
d  2 1 1
33

By using Tn  a  n 1d
 2  1  n 1  1 

 3
2 1 1  1n

33
n  10
Therefore, T10  2 .

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_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________

Example 3

a. Find the sum of all odd numbers less than 100.
b. The sum of the first 6 terms of an arithmetic progression is 96. The sum of the first 10

terms is one third of the sum of the first 20 terms. Calculate
i. The first term and common difference.
ii. The sum of the first 5 terms.

Solution:

a. Listing the numbers 1,3,5,,99

a 1

d1  3 1  2

d2  53  2

Since the sequence has a common difference, d  2 , so it is an arithmetic sequence.

By using Tn  a  n 1d
99 1 n 12

98  2n  2

n  50

Therefore, Sn  n 2a  n 1d 

2

S50  50 21  50 12
2

 2500

Or alternatively, Sn  n a  l

2

S50  50 1  99
2

 2500

b. i. Given S6  96 and S10  1
3 S20

From S6  6 2a  5d  96

2

6a 15d  96

2a  5d  32    1

From S10  1 S 2 0
3

10 2a  9d   1  20 2a 19d 
2 3 2 

10a  45d  20 a  190 d
33

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10 a  55 d  0
33

2a 11d  0   2

1 2: 16d  32

d 2
a  11

ii. S5  5 211 5 12

2

 75

Exercise

1. Show that the sequence log a , log  a3  , log  a 5  , is an arithmetic sequence.
b b 2

2. Identify each sequence as arithmetic or not. Explain your answer, and write an explicit

formula for the sequence.
a. 14,11,8,5,

b.  1 ,  3 ,  5 ,  7
2222

3. Find out whether the given sequence is an arithmetic sequence. If so, find the first term

and the difference of the arithmetic sequence.

a. Tn  5n 1
3

b. Tn  n2  3

c. Tn  n2  2n 1

4. Write down the 10th and 19th terms of the arithmetic progression

a. 8, 11, 14, . . .,

b. 8, 5, 2 . . ..

5. Write down the first five terms of the arithmetic progression with first term 2 and
common difference −5.

6. Find the 17th term of the arithmetic progression with first term 5 and common difference

2.

7. Find how many terms are between 200 and 500 are divisible by 8.
8. Is 200 any term of the sequence 3, 7,11,15,

9. The sixth term of an arithmetic progression is -10 and its tenth term is -26. Determine
the 15th term of an arithmetic progression.

10. Find an expression for the nth term of the sequence  2, 4,10, and use this formula to

find the 15th term of the sequence.

11. An arithmetic progression is given by k, 2k , k , 0,
33

a. Find the sixth term.
b. Find the nth term.
c. If the 20th term is equal to 15, find k.

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_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________

12. Find the sum of the following series
a. An arithmetic series of 25 terms where the first term is 5 and the last term is
41.
b. 3  5  7  to 60 terms.
c. 1 3.5  6  8.5 101

5

d.  r 1
r 1

13. Find the sum of the first 50 terms of the sequence 1 3 5  7 
14. Find the sum of all positive integers less than 200 which are

a. Multiples of 5.
b. Not multiples of 5
15. Find the number of terms and the sum of each of the following arithmetic sequences.
a. 7, 3,  1,  5,,  241
b. n, n 1, n  2,,3n 1
c. k, k  5, k 10,,6k
16. An arithmetic progression has 3 as its first term. Also, the sum of the first 8 terms is
twice the sum of the first 5 terms. Find the common difference.
17. The sum of the first 20 terms of an arithmetic series is identical to the sum of the first
22 terms. If the common difference is −2, find the first term.
18. An arithmetic sequence has first term 5 and common difference 1 . find the least value

2
of n such that the sum of the first n terms of the sequence exceeds 5000.
19. An arithmetic sequence has first term a and common difference d. Write in the simplest
form an expression for sum of all terms from the 16th term to the 50th term.
20. An arithmetic sequence has first term a and common difference d. the 15th term is 59
and the sum of the first 50 terms is four times the sum of the first 20 terms. Find a and
d.

6.2 Geometric Series

Definition of Geometric Sequence

A sequence T1,T2 ,T3 ,,Tn , is called a geometric sequence or geometric progression,
if there exists a nonzero constant r, called the common ratio, such that r  Tn .

Tn1

nth Term Formula of Geometric Sequence

A sequence of number in which any term can be obtained from the previous term by
multiplying the preceding term by constant nonzero real number.

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T1  a
T2  ar

T3  arr  ar2

 T4  ar2 r  ar3


Tn  ar n1

Sum to n Terms of a Geometric Series

The sum of the terms of a geometric progression gives a geometric series. If the starting
value is a and the common ratio is r then the sum of the first n terms is

 Sn
 a rn 1 , for r 1
r 1

 Sn
 a1rn , for r  1
1 r

Sum to Infinity Terms of a Geometric Series

S  a , for1  r 1
1 r

Example 1

Which of the following can be the first four terms of a geometric sequence?
a. 1,3,9, 27,
b. 1, 4, 9,16,

Solution:
a. 1,3,9, 27,

r1  3  3
1

r2  9  3
3

r3  27  3
9

Since the sequence has a common ratio, r  3 , then it is a geometric sequence.

b. 1, 4, 9,16,

r1  4  4
1

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r2  9
4

r3  16
9

Since the sequence does not have a common ratio, then it is not a geometric sequence.

Example 2

A geometric progression has 13 terms. The first and last terms are 4 and 256 respectively. Find
the common ratio and the ninth term.

Solution:

Given a  4 and T13  256 .
By using Tn  arn1

256   4 r 131

64  r12

 2 12  r12

r 2

 Therefore, T9  4 91

2

 64

Example 3

Calculate the sum of the series 9   3  1    1       1 
 3  243 

Solution:

We first must determine what sequence we have as this information has not been provided in
the question.

T1  9
T2  3
T3  1

r1  3  1
9 3

r2  1  1
3 3

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_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________

So we know we have a geometric sequence as we have a common ratio. Now we can use the
formula.

Tn  ar n1

 1  9  1 n
243  3 

 1    1 n
2187  3 

  1 7    1 n
 3  3

n7

Therefore, the sum of the series is

 Sn
 a 1 rn
1 r

91  1  7 
 3  
 

S7  1    1   674897119

 3

Example 4
The first term of a geometric progression is 4 and the sum to infinity is 8. Find the 10th term.

Solution:

Given a  4 and S  8 .

By using the formula S a
1 r

8 4
1 r

4 1r
8

r1
2

41  1 1 0 
 2  


Therefore, T1 0  1   1   1023
128

2

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_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________

Exercise

1. Identify each sequence as geometric or not. Explain your answer, and write an explicit

formula for the sequence.
a. 2,10,50, 250,

b. 4, 2,1

2. Find out whether the given sequence is a geometric sequence. If so, find the first term
and the common ratio of the geometric sequence.

a. Tn  5

b. Tn  3n

c. Tn   2 1n
3

3. Find the sixth and nth terms of the following geometric sequences.
a. 4,  8,16,  32,

b. 2, 2 , 2 , 2 ,
3 9 27

c. 12, 6,3, 3 ,
2

4. The first three terms of a geometric sequence are x  2, x and 2x  3 . if these three

terms are positive, find

a. The value of x.
b. The 5th term.

5. The first term of a geometric sequence is 2  2 and the second term is 2  2 . Find,
in the simplest form, the third and fourth terms.

6. The first term of a geometric sequence is 24 and the fourth term is -81. Find the
common ratio and the second term.

7. The second and fifth terms of a geometric progression are 12 and 324 respectively.
Find the 7th term.

8. Find the sum of the geometric series 2 + 6 + 18 + 54 + … where there are 6 terms in the
series.

9. Find the sum of the geometric series 8−4 + 2−1 + … where there are 5 terms in the
series.

10. The first term of a geometric sequence is 7, the last term is 448 and the sum of all these
terms is 889. find the common ratio.

11. The sum of the first three terms of a geometric sequence is 336, and the sum of the 5th,
6th and 7th term is 21. Find the possible values of the 3rd term.

5

12. Find the sum to of the geometric series  3r .
r0

13. Calculate the sum to infinity of the following series.

a. 1  1  1  
6 36

b. 0.1 0.01  0.001  0.0001 
c. a  ab2  ab4  ab6  

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14. Find the sum to infinity of the geometric sequence with first term 3 and common ratio
1.
2

15. The sum to infinity of a geometric sequence is four times the first term. Find the
common ratio.

16. The sum to infinity of a geometric sequence is twice the sum of the first two terms.
Find possible values of the common ratio.

17. The sum to infinity of a geometric series is 12 and the sum of the first three terms is
76 . Find
9
a. the common ratio.
b. the least value of n such that the sum of the first n terms is greater than 99%
of the sum to infinity.

6.3 Applications of Arithmetic and Geometric Series

Example1

A lecture hall has 20 rows of seats in the first row, 28 in the second, 31 in the third and so on.
Find the number of seats in the 17th row of the lecture hall.

Solution:
T1  25
T2  28
T3  31

d1  28  25  3
d2  31 28  3

So we know we have an arithmetic sequence as we have a common difference. Now we can
use the formula.

Tn  a  n 1d
T17  25  17 13  73

The number of seats in the 17th row of the lecture hall is 73.

Example 2

A tired dog is trying to find its way home. It travels 900 metres in the first hour, 720 metres in
the second hour and 576 metres in the third hour. If the dog travels up to sixth hours, determine
his total travels distance.

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Solution:
We first must determine what sequence we have as this information has not been provided in
the question.

T1  900
T2  720
T3  576

r1  720  0.8
900

r2  576  0.8
720

So we know we have a geometric sequence as we have a common ratio. Now we can use the
formula.

 S6
 900 1  0.86 =3320.35 m

1  0.8

The dog’s total travels distance is 3.32 km.

Example 3

A photocopier was purchased for $13,000 in 2014. The photocopier decreases in value by 20%
of the previous year’s value.

a. What is an expression for the value of the photocopier, Vn , after n years?
b. What is the value of the photocopier after three years?

Solution:
a. V1  13000

V2  V10.8  130000.8
V3  V2 0.8  130000.80.8  1300000.82
V4  V3 0.8  130000.82 0.8  1300000.83

Therefore, the value of the photocopier
Vn  arn1

 130000.8n1

b. Use the expression found in part a) and substitute the value of n.

Vn  130000.8n1
V3  130000.831

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 $8320
The value of the photocopier after three years is $8,320.

Exercise

1. The local football team won the championship several years ago, and since then, ticket
prices have been increasing $20 per year. The year they won the championship, tickets
were $50. Write a recursive formula for a sequence that models ticket prices. Is the
sequence arithmetic or geometric?

2. A radioactive substance decreases in the amount of grams by one-third each year. If the
starting amount of the substance in a rock is 1.452 g. Write a recursive formula for a
sequence that models the amount of the substance left after the end of each year. Is the
sequence arithmetic or geometric?

3. A man’s initial annual salary was RM 600 and increased by RM45 a year. How much
does he expect to earn after 15 years?

4. The price of a new car is RM 42000. If the value of the car depreciates at a rate of 10%
each year, find the value of the car after 12 years.

5. A ball is released from rest at a height H meters above a horizontal surface and rebounds

to a height of 3 H meters. If the ball is released from rest at a height of 10 meters, find
4

the height of the ball at 10th rebound.

NFTF Exercises

6.1 Arithmetic Series

1. Do the numbers 2, 6, 10, 12, 16... form an arithmetic progression? Explain why.

Ans: No

2. Identify each sequence as arithmetic or not. Explain your answer, and write an explicit

formula for the sequence.

a. 14, 21, 28,35, Ans: Yes Tn  7  7n

b. x  4, x  8, x  12, x  16 Ans: Yes Tn  x  4n

3. Find out whether the given sequence is an arithmetic sequence. If so, find the first term

and the difference of the arithmetic sequence.

a. Tn  3 Ans: Yes

b. Tn  5  2n Ans:Yes

c. Tn  1 Ans: No
n

4. The first term of an arithmetic sequence is 4 and the tenth term is 67. What is the

common difference? Ans: 7

5. What is the thirty-second term of the arithmetic sequence 12,  7,  2,3,?

Ans: 143

6. Given the tenth term and the fifth of an arithmetic progression are 15 and 5 respectively.

Find the first term. Ans: -3

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7. An arithmetic progression has first term log 2 27 and common difference log 2 x .

 a. Show that the fourth term can be written as log 2 27x3 .

b. Given that the fourth term is 6, find the exact value of x. Ans: 4 .
3

8. Find the number of terms that is multiples of 9, between 30 and 901. Ans: 97

9. How many terms of the arithmetic sequence 2, 8, 14, 20, ... are required to give a sum

of 660? Ans: 15

10. The eleventh term of an arithmetic sequence is 30 and the sum of the first eleven terms

is 55. What is the common difference? Ans: 5

11. Find the sum of the eleventh to twentieth terms (inclusive) of the arithmetic sequence

7,12,17, 22, Ans: 795

12. Let an be an arithmetic progression, for which a1  a2  a3  102 and a1  15. Find

a10 Ans: 186

13. Let an  be an arithmetic progression. If a3  a8  a10  a16  a18  a23  126 , find the

sum of the first 25 terms of an Ans: 525

14. Find the sum of the first three elements of an arithmetic progression, for a1  a5  22

and a8  a5  6 . Ans: 27

15. We put a few numbers between numbers 12 and 48 so that all the numbers together now

form the increasing finite arithmetic sequence. The sum of all entered numbers is 330.

What is the difference of the arithmetic sequence ? Ans: 3

6.2 Geometric Series

16. Identify each sequence as geometric or not. Explain your answer, and write an explicit

formula for the sequence.

a. 10,1, 1 , Ans: Yes Tn  10010n
10

b. 4, 40, 400, 4000, Ans: Yes Tn  4 10n
10

c. 49, 7,1, 1 , 1 , Ans: Yes Tn  4971n
7 49

17. Find out whether the given sequence is a geometric sequence. If so, find the first term

and the common ratio of the geometric sequence.

a. Tn  nn Ans; No

b. Tn  n  12 Ans: No

2

c. Tn  4  31n Ans: Yes

18. Write down the eleventh term of the geometric progression 1 , 1 , 1 , Ans: 1
3 6 12 3072

19. Find the number of terms in the geometric progression 6, 12, 24, ..., 1536 Ans: 9

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_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________

20. The 5th term of a geometric progression is  48 and the 7th term is 12 . Find the values

of the common ratio and the first term. Ans: a  768 r 1
2

21. The 12th term of a geometric sequence is 2 and the 15th term is 54. Find the common

ratio and the first term.

Ans: r  3 a 2
177147

22. The first term of a geometric sequence is 5 and the sixth term is 160. What is the

common ratio? Ans: 2

23. Find the sum specified for each of the following of the geometric series

a. 5 10  20  find S10. Ans: 5115

b. 1 3  9  find S6 . Ans: -182

c. 2  6 18 1458 . Ans: 1094

24. How many terms of the geometric sequence 2, 8, 32, 128,... are required to give a sum

of 174,762? Ans: 9

25. The sum of the first and the third term of a geometric sequence is 15. The sum of the

first three terms of this sequence is 21. Determine the first term and the quotient of this

sequence.

Ans: a = 3; r = 2 or a = 12; r = 1/2

26. Find the sum of each of the geometric series  2  1  1  1 Ans: 52429
28 37268 60536

27. Find the sum of

10 Ans: 1023

a. 3 2 n1
n 1

b.   1 r Ans: 1
r 1  3  2

 Ans: 9
2
c. 2 3r2
r 0

28. A geometric progression has first term log 2 27 and common ratio log 2 y .

a. Find the sum to infinity. Ans: log 2 27
 2 
log 2 y

b. Find the exact value of y for which the sum to infinity of the geometric

progression is 3. Ans: 2
3

29. The sum of the first three terms of a geometric progression is  3 and the sum to
2

infinity is 3 . If the geometric progression has a positive common ratio, r. Find r and
25

the first term. Ans: r  3 a   3
26

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_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________

6.3 Applications of Arithmetic and Geometric Series

30. In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers

than the year before, so that the shop sold 160 computers in year 2001, 170 computers

in year 2002 and so on.

a. Show that the shop sold 220 computers in year 2007.

b. Calculate the total number of computers the shop sold in 2000 to 2013 inclusive.

Ans: b) 3010

31. A company, which is making 200 mobile phones each week, plans to increase its

production. The number of mobile phones produced is to be increased by 20 each week

from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing

600 in week X.

a. Find the value of X. Ans: 21

The company then plans to continue to make 600 mobile phones each week.

b. Find the total number of mobile phones that will be made in the first 52 weeks

starting from and including week 1. Ans: 27000

32. Lewis played a game of space invaders. He scored points for each spaceship that he

captured. Lewis scored 140 points for capturing his first spaceship. He scored 160 points

for capturing his second spaceship, 180 points for capturing his third spaceship, and so

on. The number of points scored for capturing each successive spaceship formed an

arithmetic sequence.
a. Find the number of points that Lewis scored for capturing his 20th spaceship.

Ans: 520

b. Find the total number of points Lewis scored for capturing his first 20

spaceships. Ans: 6600

33. Lengths of the sides of a right-angled triangle are three consecutive terms of an

arithmetic sequence. Calculate the length of the sides, if you know:

a. perimeter of the triangle is 72 cm

Ans: a = 18 cm; b = 24 cm; c = 30 cm

b. area of the triangle is 54 cm2

Ans: a = 9 cm; b = 12 cm; c = 15 cm

34. The student needs to study 313 pages of math textbook for an exam he has in 14 days.

If the first day he studied 35 pages and each following day he studied two pages less

than the previous day, how many pages will he have to handle during the day before the

exam ? Ans: 14

35. Dimensions of a cuboid are consecutive terms of a geometric sequence. The volume of
the cuboid is 216 cm3 and the surface of the cuboid is 312 cm2. Determine the

dimensions of the cuboid.

Ans: a = 2 cm; b = 6 cm; c = 18 cm

36. The rabbit population in a Victorian town was estimated to be 320,000 in 2012.

Scientists believe that this will increase by two percent each year.

a. What will the rabbit population be in 2015? Round your answer to the nearest

decimal place. Ans: 339,587

b. In which year will the rabbit population reach 400,000? Ans: 2023

37. A mine worker discovers an ore sample containing 500mg of radioactive materials. It is

discovered that the radioactive material has a half-life of 2 days. Find the amount of
radioactive material in the sample in the beginning of the 14th day. Ans: 7.8125mg

16

_______CHAPTER 6: ARITHMETIC SERIES AND GEOMETRIC SERIES_________

38. An island has a population of 1500 and is growing at a rate of 3% per year. What will

the population be after 6 years? Ans: 1791

39. A species of fish is going to extinct. There are 1700 fish. If the species is decreasing at

a rate of 5% per year.

a. How many fish will there be after 4 years? Ans: 1385

b. How long will it take for half the fish to die? Ans: 13.51 years

40. A bob of a pendulum swings through an arc of 50 cm on its first swing. Each successive

swing is 90% of the length of the previous swing. Find the total distance the bob travels

before coming to rest. Ans: 5 m

17