2. Chapter 8: Further Topics in Algebra
8.1 Sequences and Series
8.2 Arithmetic Sequences and Series
8.3 Geometric Sequences and Series
8.4 The Binomial Theorem
8.5 Mathematical Induction
8.6 Counting Theory
8.7 Probability

3. 8.2 Arithmetic Sequences and Series
An arithmetic sequence is a sequence in which each term is obtained by adding a fixed number to the previous term.
5, 9, 13, 17 … is an example of an arithmetic sequence since 4 is added to each term to get the next term.
The fixed number added is called the common difference.

4. 8.2 Finding a Common Difference
Example Find the common difference d for
the arithmetic sequence –9, –7, –5, –3, –1, …
Solution d can be found by choosing any two
consecutive terms and subtracting the first
from the second:
d = –5 – (–7) = 2 .

5. 8.2 Arithmetic Sequences and Series
nth Term of an Arithmetic Sequence
In an arithmetic sequence with first term a1 and common difference d, the nth term an, is given by

6. 8.2 Finding Terms of an Arithmetic Sequence
Example Find a13 and an for the arithmetic
sequence –3, 1, 5, 9, …
Solution Here a1= –3 and d = 1 – (–3) = 4. Using
n=13,
In general

7. 8.2 Find the nth term from a Graph
Example Find a formula for the nth term of the sequence graphed below.

8. 8.2 Find the nth term from a Graph
Solution The equation of the dashed line shown
Below is y = –.5x +4.
The sequence is given by an = –.5n +4 for
n = 1, 2, 3, 4, 5, 6 .

9. 8.2 Arithmetic Sequences and Series
Sum of the First n Terms of an Arithmetic Sequence
If an arithmetic sequence has first term a1 and common difference d, the sum of the first n terms is given by or

10. 8.2 Using The Sum Formulas
Example Find the sum of the first 60 positive
integers.
Solution The sequence is 1, 2, 3, …, 60 so a1 = 1
and a60 = 60. The desired sum is

11. 8.2 Using Summation Notation
Example Evaluate the sum .
Solution The sum contains the terms of an
arithmetic sequence having a1 = 4(1) + 8 = 12 and
a10 = 4(10) + 8 = 48. Thus,