Series and Sequences
An infinite sequence is an unending list of
numbers that follow a pattern. The terms of
the sequence are written a1, a2, a3,...,an,...
If the list ends, we call it a finite sequence.
Ex. Write the first four terms of the sequence:
a) an = 3n – 2
b) an = 3 + (-1)n
Ex. Write the nfirst four terms of the sequence
1

an 
2n1
Ex. Write an expression for an:
a) 1, 3, 5, 7,...
b) 2, -5, 10, -17,...
A sequence is recursive if each term is
defined by one or more previous terms
Ex. The Fibonacci sequence is defined as
a0 = 1, a1 = 1, ak = ak – 1 + ak – 2. Write the
first six terms.
Ex. Find the first five terms of the recursive
sequence defined by a1 = 3, ak = 2ak – 1 – 5
If n is a positive integer, n factorial is
defined as
n! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ ... ∙ n
As a special case, 0! = 1.
Keep in mind that parentheses matter:
2n! = 2 ∙ n! = 2(1 ∙ 2 ∙ 3 ∙ 4 ∙ ... ∙ n)
(2n)! = 1 ∙ 2 ∙ 3 ∙ 4 ∙ ... ∙ 2n
Ex. Write nthe first five terms of the sequence
2
an 
n!
Ex. Evaluate the factorial
8!
a)
2! 6!
2! 6!
b)
3! 5!
n  1!

c)
 n  1!
The Greek letter sigma (Σ) can be used to
show the sum
of
many
terms
n
a
a

a

a

.
.
.

a

i
1
2
3
n
i

1
i is called the index of the summation
n is the upper limit of the summation
1 is the lower limit of the summation
Ex. Find the sum
5
a)  3 i
i 1
1k 
b) 
6
2
k3
8
c) 
i0
1
i!
Consider the infinite sequence a1, a2, a3,..., ai,...
The sum of the first n terms is called the nth partial
sum of the sequence, and is denoted
n
a

a

a

a

.
.
.

a

S

i
i

1
1 2 3
n n
The sum of all the terms of the infinite sequence is
called an infinite series, and is denoted

a

a

a

a

.
.
.

a

.
.
.

i
i

1
1 2 3
i

3
Ex. Use the first 3 partial sums to evaluate the sum  i
i 1 10
Practice Problems
Section 8.1
Problems 1, 17, 37, 51, 59, 67, 73, 99
Arithmetic Sequences and Series
A sequence is arithmetic if the difference of
two consecutive terms is the same.
an + 1 – an = d for any positive integer n
The number d is called the common difference
Ex. Find the first 4 terms of the arithmetic
sequence.
a) an = 4n + 3
b) an = 7 – 5n
c) a
1

n

3


n 4
To find the nth term of an arithmetic
sequence, we use the formula
an = a1 + d(n – 1)
where a1 is the first term and d is the
common difference
Ex. Find the nth term of the sequence
2, 5, 8, 11, 14,...
Ex. The fourth term of an arithmetic sequence
is 20 and the 13th term is 65. Find the nth
term.
Ex. Find the 9th term of the arithmetic
sequence that starts with 2 and 9.
To find the sum of a finite arithmetic
sequence with n terms, we use the formula
n
S n   a1  an 
2
Ex. Find the sum of the first 10 odd numbers.
Ex. Find the sum of the integers from 1 to 100
Ex. Find the 150th partial sum of the
arithmetic sequence
5, 16, 27, 38, 49,...
100
Ex. Find the sum
 7n
n  51
Ex. In a golf tournament, 16 golfers win cash prizes. First
place gets $1000, second place gets $950, third place
gets $900, and so on. What is the total amount of prize
money?
Practice Problems
Section 8.2
Problems 1, 21, 37, 45, 63, 65, 69, 89