CHAPTER 6
Sequences and Series
Around 1260 AD, the Kurdish historian Ibn Khallik¯an recorded the
following story about Sissa ibn Dahir and a chess game against the
Indian King Shihram. (The story is also told in the Legend of the
Ambalappuzha Paal Payasam, where the Lord Krishna takes the place
of Sissa ibn Dahir, and they play a game of chess with the prize of rice
grains rather than wheat.) King Shihram was a tyrant king, and his
subject Sissa ibn Dahir wanted to teach him how important all of his
people were. He invented the game of chess for the king, and the king
was greatly impressed. He insisted on Sissa ibn Dahir naming his
reward, and the wise man asked for one grain of wheat for the first
square, two grains of wheat for the second square, four grains of wheat
for the third square, and so on, doubling the wheat on each successive
square on the board.
The king laughed at first and agreed, for there was so little grain on the
first few squares. By halfway he was surprised at the amount of grain
being paid, and soon he realized his great error: that he owed more
grain than there was in the world.
a. How can we describe the number of grains of wheat for
each square?
b. What expression gives the number of grains of wheat for
the nth square?
• Find the total number of grains of wheat that the king owed.
In mathematics it is important that we can:
recognize a pattern in a set of numbers,
describe the pattern in words, and
continue the pattern
A number sequence is an ordered list of numbers
defined by a rule.
The numbers in the sequence are said to be its
members or its terms.
A sequence which continues forever is called an infinite
sequence.
A sequence which terminates is called a finite
sequence.
4. Write down the first four terms of the sequence if you
start with
a. 45 and subtract 6 each time
b. 96 and divide by 2 each time.
4. Write down the first four terms of the sequence if you
start with
a. 45 and subtract 6 each time
45, 39, 33, 27
b. 96 and divide by 2 each time.
96, 48, 24, 12
5. For each of the following write a description of the
sequence and find the next 2 terms:
a. 96, 89, 82, 75, ....
b. 50 000, 10 000, 2000, 400, ....
5. For each of the following write a description of
the sequence and find the next 2 terms:
a. 96, 89, 82, 75, ....
start with 96 and subtract 7 each time
68, 61
b. 50 000, 10 000, 2000, 400, ....
start with 50,000 and multiply 1/5 each
time
80, 16
Sequences may be defined in one of the following
ways:
• listing all terms (of a finite sequence)
• listing the first few terms and assuming that the
pattern represented continues indefinitely
• giving a description in words
• using a formula which represents the general
term or nth term.
The general term or nth term of a sequence is
represented by a symbol with a subscript, for
example
un, Tn, tn, or An. The general term is defined
for n = 1, 2, 3, 4, ....
{un} represents the sequence that can be
generated by using un as the nth term.
8.Consider the sequence defined by un = 2n+ 5.
Find the first four terms of the sequence
8.Consider the sequence defined by un = 2n+ 5.
Find the first four terms of the sequence.
n=1
n=2
n=3
n=4
u1 = 2(1) + 5 = 7
u2 = 2(2) + 5 = 9
u3 = 2(3) + 5 =11
u4 = 2(4) + 5 = 13
1
9. Evaluate the first five terms of the sequence, 6   
2
n
1
9. Evaluate the first five terms of the sequence, 6   
2
1
n 1
3
1
6   
1
2
2
n2
6
3
1
6    
4 2
2
3
n3
6 3
1
6    
8
4
2
4
n4
6
3
1
6   

16
8
2
5
n5
6
3
1
6   

32 16
2
n
10. Evaluate the first five terms of the sequence, .
15    2 
n
10. Evaluate the first five terms of the sequence. 15    2 n
n 1
15    2   17
n2
15    2   11
n3
15    2   23
n4
15    2    1
n5
15    2   47
1
2
3
4
5
11. An arithmetic sequence is a sequence in
which each term differs from the previous one by
the same fixed number.
It can also be referred to as an arithmetic
progression.
12. {un} is arithmetic  un+1 – un = d for all positive
integers n where constant called the common
difference.
13.For an arithmetic sequence with first term u1
and common difference d the general term or nth
term is
un = u1 + (n – 1)d.
14. Find the 10th term of each of the following
arithmetic sequences 101, 97, 93, 89, ....
14. Find the 10th term of each of the following
arithmetic sequences 101, 97, 93, 89, ....
un = u1 + (n – 1)d.
u10 = 101 + (10 – 1)(-4) = 65
15. Find the 15th term of each of the following
arithmetic sequences: a, a + d, a+ 2d, a + 3d, ....
15. Find the 15th term of each of the following
arithmetic sequences: a, a + d, a+ 2d, a + 3d, ....
u15 = a + (15 – 1)d = a + 14d
16. Consider the sequence 87, 83, 79, 75, 71, ....
a. Show that the sequence is arithmetic.
b. Find the formula for its general term.
c. Find the 40th term.
d. Which term of the sequence is -297?
16. Consider the sequence 87, 83, 79, 75, 71, ....
a. Show that the sequence is arithmetic.
83 – 87 = 79 – 83 = 75 – 79 = 71 – 75 = -4
b. Find the formula for its general term.
un = 87 + (n – 1)(-4) = 87 – 4n + 4 = 91 – 4n
c. Find the 40th term.
un = 91 – 4(40) = -69
d. Which term of the sequence is -297?
-297 = 91 – 4n

n = 97
17. A sequence is defined by .
un 
71  7 n
2
a. Prove that the sequence is arithmetic.
b. Find u1 and d.
c. Find u75.
d. For what values of n are the terms of the
sequence less than -200?
18. Find k given the consecutive arithmetic terms:
a. k +1, 2k + 1, 13
b. k – 1, 2k + 3, 7 – k
19. Find the general term un for an arithmetic
sequence with:
a. u7 = 41 and u13 = 77
b. seventh term 1 and fifteenth term -39
20. A luxury car manufacturer sets up a factory for a new
model. In the first month only 5 cars are produced. After
this, 13 cars are assembled every month.
a. List the total number of cars that have been made in the
factory by the end of each of the first six months.
b. Explain why the total number of cars made after n
months forms an arithmetic sequence.
c. How many cars are made in the first year?
d. How long is it until the 250th car is manufactured?
20. A luxury car manufacturer sets up a factory for a new model.
In the first month only 5 cars are produced. After this, 13 cars are
assembled every month.
a. List the total number of cars that have been made in the
factory by the end of each of the first six months.
5, 18, 31, 44, 57, 70
b. Explain why the total number of cars made after n months
forms an arithmetic sequence.
constant difference, d = 13
c. How many cars are made in the first year?
u12 = 5 + (12 – 1)(13) = 148
d. How long is it until the 250th car is manufactured?
250 = 13n – 8  n = 19.84 20
21. A sequence is geometric if each term can be obtained
from the previous one by multiplying by the same non-zero
constant.
A geometric sequence is also referred to as a geometric
progression
{un } is geometric 
u n 1
un
 r
for all positive integers
n constant called the
common ratio.
For a geometric sequence with first term u1 and
common ratio r, the general term or nth term is
u n = u 1r n – 1.
24. For the geometric sequence with first two terms given,
find b and c: 2, 6, b, c, ....
24. For the geometric sequence with first two terms given,
find b and c: 2, 6, b, c, ....
common ratio = 6/2 = 3
b = 6 • 3 = 18
c = 18 • 3 = 54
25. Find the 9th term in each of the following geometric
sequences: 12, 18, 27, ....
26. a. Show that the sequence 12, -6, 3, -3/2 , .... is
geometric.
26. a. Show that the sequence 12, -6, 3, -3/2 , .... is
geometric.
3
6
3
1
2

 

12
6
3
2
b. Find un and hence write the 13th term as a rational
number.
26. b. Find un and hence write the 13th term as a rational
number.
u 13
 1
 12   
 2
13  1

3
1024
27. Find k given that the following are consecutive terms of
a geometric sequence
k, k + 8, 9k
28. Find the general term un of the geometric sequence
which has:
u3 = 8 and u6 = -1
A nest of ants initially contains 500 individuals.
The population is increasing by 12% each week.
a. How many ants will there be after:
i 10 weeks
ii 20 weeks?
b. Use technology to find how many weeks it will take for
the ant population to reach 2000.
We can calculate the value of a compounding investment
using the formula
un+1 = u1 x r n
where u1 = initial investment
r = growth multiplier for each period
n = number of compounding periods
un+1 = amount after n compounding periods.
31 a. What will an investment of $3000 at 10% p.a.
compound interest amount to after 3 years?
b. How much of this is interest?
32. How much compound interest is earned by investing
$80 000 at 9% p.a. if the investment is over a 3 year
period?
33. What initial investment is required to produce a
maturing amount of $15 000 in 60 months’ time given a
guaranteed fixed interest rate of 5.5% p.a. compounded
annually?
A series is the sum of the terms of a sequence.
For the finite sequence {un} with n terms, the
corresponding series is u1 + u2 + u3 + …+ un.
The sum of this series is Sn = u1+u2+u3+…+un and
this will always be a finite real number.
For the infinite sequence {un} , the corresponding series is
u1 + u2 + u3 + … + un + …
In many cases, the sum of an infinite series cannot
be calculated. In some cases, however, it does converge to
a finite number.
Sigma Notation
n
u1 + u2 + u3 + … + un 
u
k 1
k
Properties of Sigma Notation
39. For the following sequences:
i. write down an expression for Sn
ii. find S5.
a. 3, 11, 19, 27, ....
b. 12, 6, 3, 1.5, ....
39. For the following sequences:
i. write down an expression for Sn
ii. find S5.
a. 3, 11, 19, 27, ....

 8 k  5 
k 1
5
 8 k  5   3  11  19  27  35  95
k 1
40. Expand and evaluate:
6
a . ( k  1)
k 1
7
b . k ( k  1 )
k 1
40. Expand and evaluate:
6
a . ( k  1)  2  3  4  5  6  7  27
k 1
7
b . k ( k  1)  2  6  12  20  30  42  56  168
k 1
41. For un = 3n – 1, write u1 + u2 + u3 + . . . + u20 using
sigma notation and evaluate the sum.
41. For un = 3n – 1, write u1 + u2 + u3 + . . . + u20 using
sigma notation and evaluate the sum.
20
 3 k  1 
k 1
2  5  8  11  14  17  20  23  26  29  32  35  38  41  44 
47  50  53  56  59  610
42. An arithmetic series is the sum of the terms of an
arithmetic sequence.
44. Find the sum of:
3 + 7 + 11+ 15 + . . . to 20 terms
50 + 48.5 + 47 + 45.5. . . to 80 terms
44. Find the sum of:
3 + 7 + 11+ 15 + . . . to 20 terms
Sn 
S 20 
n
2
( 2 u 1  ( n  1) d )
20
2
( 2 ( 3 )  ( 20  1)( 4 ))  820
50 + 48.5 + 47 + 45.5. . . to 80 terms
Sn 
n
S 20 
80
2
( 2 u 1  ( n  1) d )
2
( 2 ( 50 )  ( 80  1)(  1 . 5 ))   740
45. Find the sum of: 5 + 8 + 11+ 14 + . . . + 101
45. Find the sum of: 5 + 8 + 11+ 14 + . . . + 101
u n  u 1  ( n  1) d
101  5  ( n  1)( 3 )
99  3 n
33  n
Sn 
n
S 33 
33
2
u 1  u n 
( 5  101 )
2
S 33  1749
46. Evaluate the arithmetic series:
46. Evaluate the arithmetic series:
k 3
 2 

k 1 
20
k 1
1 3
S 20
S 20  135
2
2
k 2
23
2
d 
1
2
u1  2
u 20 
23
2

20 
23 

2 

2 
2 
5
2
47. A soccer stadium has 25 sections of seating. Each
section has 44 rows of seats, with 22 seats in the first row,
23 in the second row, 24 in the third row, and so on. How
many seats are there in:
a. row 44 of one section
b. each section
c. the whole stadium?
48. Three consecutive terms of an arithmetic sequence
have a sum of 12 and a product of -80. Find the terms.
Hint: Let the terms be x – d, x, and x + d.
49. A geometric series is the sum of the terms of a
geometric sequence.
51. Find the sum of the following series:
12+ 6 + 3 + 1.5 + . . . to 10 terms
1
1
2

1
2

1
2 2

. . . 20 terms
52. Find a formula for Sn, the sum of the first n terms of the
series:
0.9 + 0.09 + 0.009 + 0.0009 + . . .
53. A geometric sequence has partial sums
S1 = 3 and S2 = 4.
a. State the first term u1.
b. Calculate the common ratio r.
c. Calculate the fifth term u5 of the series.
54. Evaluate these geometric series:
10
 32
k 1
k 1
55.
56. Find the sum of each of the following infinite geometric
series:
a. 18 + 12 + 8 + 16/3 + . . .
b. 18.9 – 6.3 + 2.1 – 0.7 + . . .
56. Find the sum of each of the following infinite geometric
series:
a. 18 + 12 + 8 + 16/3 + . . .
b. 18.9 – 6.3 + 2.1 – 0.7 + . . .
57. Find the following

 2
 6  5 

k 0 
k