Arithmetic Sequences

A simple way to generate a sequence is to start with a number a, and add to it a fixed

constant d, over and over again. This type of sequence is called an arithmetic sequence.

Definition: An arithmetic sequence is a sequence of the form

a, a + d, a + 2d, a + 3d, a + 4d, …

The number a is the first term, and d is the common difference of the

sequence. The nth term of an arithmetic sequence is given by

an = a + (n – 1)d

The number d is called the common difference because any two consecutive terms of an

arithmetic sequence differ by d, and it is found by subtracting any pair of terms an and

an+1. That is

(d = an+1 – an)

Is the Sequence Arithmetic?

Example 1: Determine whether or not the sequence is arithmetic. If it is arithmetic, find

the common difference.

(a) 2, 5, 8, 11, …

(b) 1, 2, 3, 5, 8, …

Solution (a): In order for a sequence to be arithmetic, the differences between

each pair of adjacent terms should be the same. If the differences

are all the same, then d, the common difference, is that value.

Step 1: First, calculate the difference between each pair of adjacent

terms.

5 – 2 = 3

8 – 5 = 3

11 – 8 = 3

Step 2: Now, compare the differences. Since each pair of adjacent terms

has the same difference 3, the sequence is arithmetic and the

common difference d = 3.