College Algebra 9th txtbk

520 CHAPTER 8 SEQUENCES, INDUCTION, AND PROBABILITY
8-3 Arithmetic and Geometric Sequences
Z Arithmetic and Geometric Sequences
Z Developing nth-Term Formulas
Z Developing Sum Formulas for Finite Arithmetic Series
Z Developing Sum Formulas for Finite Geometric Series
Z Developing a Sum Formula for Infinite Geometric Series
For most sequences, it is difficult to add up an arbitrary number of terms of the sequence
without adding the terms one at a time. In this section, we will study two special types
of sequences, arithmetic sequences and geometric sequences. One of the things that
make them special is that we can develop formulas for the sum of the corresponding
series.
Z Arithmetic and Geometric Sequences
Consider the sequence defined by the general term a n 5 2(n 1), n 1. The first
five terms are 5, 7, 9, 11, and 13. It’s not hard to see that after starting at 5, every term
is obtained by adding 2 to the previous term. This is an example of an arithmetic
sequence.
Z DEFINITION 1 Arithmetic Sequence
A sequence
a 1 , a 2 , a 3 , . . . , a n , . . .
is called an arithmetic sequence, or arithmetic progression, if there exists a
constant d, called the common difference, such that
That is,
a n a n1 d
a n a n1 d for every n 1
In short, a sequence is arithmetic when every term is obtained by adding some fixed
number to the previous term. This fixed number is called the common difference, and is
usually represented by the letter d.
Now consider the sequence with general term a n 5(2) n1 . The first five terms are
5, 10, 20, 40, and 80. It also starts at 5, but this time every term is obtained by multiplying
the previous term by 2. This is an example of a geometric sequence.