Section 8.1 – Sequences and Summation Notation
A sequence is a function whose domain is the positive integers. (n = 1, 2, 3, …)
The terms of a sequence are the numbers in the sequence and are represented by:
a1, a2, a3, …, an, …
The nth term of a sequence is the general (generic) term of the sequence.
Examples:
1. Write the first five terms of: {n2 + 1}
2. Write the first four terms of: {(–1)n+1 5n}
The factorial symbol n! is defined as:
Examples:
n! = n(n – 1)(n – 2) ∙∙∙3∙2∙1
and 0! = 1
a) 4! =
b)
8!
=
5!
1
n
Summation Notation:
a
i 1
i
 a1  a2 
 an
Examples: Find the following sums:
6
1.
 i!
i 1
4
2.
1
i
i 1
2
3
3.
 (4
i
 1)
i1
4
4.
 (1) 3
i
i
i1
2
Section 8.2 – Arithmetic Sequences
An arithmetic sequence is one in which each term after the first is obtained by adding a non-zero
constant to the preceding term. The constant is called the common difference.
Examples: Write out the first 4 terms of each arithmetic sequence.
a) {5 – 2n}
Consider:
Find the 10th term of:
b) a1 = 8, d = 3
7, 12, 17, 22, …..
To get to the 10th term in the sequence, we started with the first term, _______, and added the
number ______ ______ times.
The nth General Term of an Arithmetic Sequence is given by:
an  a1  (n  1)d
a1 = first term
d = common difference
Examples:
1. Find a formula for the nth term if a1 = 5 and d = –6.
Then, use that formula to find a23.
2. Find a formula for the nth term of: 5, 15, 25, ….
Then, use that formula to find the 42nd term.
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Consider:
Find the sum of the first twelve terms of:
4, 6, 8, …
The Sum of n Terms of an Arithmetic Sequence is given by:
Sn =
n
 a1  an 
2
Examples:
1. Find the sum of the first fifteen terms of:
8, 11, 14, ….
2. Find the sum of the first twenty terms of:
10, 9.5, 9, 8.5, …
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Example: Write out the first three terms and the last term, then find the sum
(using the formula from this section):
200
  6i  4 
i 1
Example: A display of soup cans has a total of 15 rows. The bottom row has 50 cans, and each
successive row has 3 less cans than the prior row.
a) How many soup cans are on the top row?
b) How many total soup cans are in the display?
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Section 8.3 – Geometric Sequences and Series
A geometric sequence is one in which each term after the first is obtained by multiplying the
preceding term by a fixed non-zero constant. The constant is called the common ratio.
Examples: Write out the first 4 terms of the geometric sequence.
{3n – 1}
Consider:
Find the 6th term of: 3, 12, 48, …..
To get to the 6th term in the sequence, we started with the first term, ______, and multiplied by
the number ______ ______ times.
The nth term of a Geometric Sequence is given by:
an  a1  r 
n 1
(r ≠ 0)
a1 = first term
r = common ratio
Examples:
1. Find a formula for the nth term if a1 = 2 and r = 5. Then, use that formula to find the 8th term.
2. Find a formula for the nth term of: 5, 15, 45, ….. Then, use that formula to find the 10th term.
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3. Find the 6th term of:
1 1 1
, ,
, ...
3 9 27
The sum of n terms of a Geometric Sequence is given by:
Sn =
a1 1  r n 
1  r 
(r ≠ 0, 1)
Examples:
1. Find the sum of the first twelve terms of:
3, 6, 12, 24, …
2. Find the sum of the first seven terms of:
0.01, 0.1, 1, 10, …..
Example: Find the indicated sum (using the formula from this section):
 52 
10
i
i1
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Geometric Series
An infinite sum of the form: a + a r + a r2 + a r3 + ….. + a rn–1 + …..

Infinite Geometric Series and is denoted:
 a r 
i 1
is called an
i 1
1
Examples: Write out some of the terms in the following infinite series:

1.
8 
i 1
1 i 1
3

2.
 4  2
i 1
=
=
i 1
The Sum of an Infinite Series is given by:
S 
a1
1  r 
where |r| < 1
Examples: Find the sum:
16 16

 ...
3
9
1.
16 
2.
5  10  20  40  ...

3.
 4  2
i 1
i 1
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Example: You are offered a job that pays $30,000 for the first year with an annual increase of 5% per
year beginning in the second year. That is, beginning in year 2, your salary will be 1.05 times what it
was in the previous year. What can you expect to earn in your 10th year on the job?
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