An arithmetic sequence is a set of ordered numbers in which the difference between any two successive terms is the same, or the next term in the sequence is found by adding the same number.

## What is an Arithmetic Sequence?

Any sequence where the difference between each consecutive term is constant can be classified as an arithmetic sequence.

The following list of numbers represents an arithmetic sequence:

$1,6,11,16$  as each term represents an increase of 5 units.

$6,3,0,-3,-6$  as each term represents a decrease of 3 units.

To establish whether a sequence is arithmetic we must calculate the difference between each of the terms by subtracting the first term from the second term, the second term by the third term and so on.

$d=t_2-t_1=t_3-t_2=t_4-t_3=...$

### Example 1

Which of the following are arithmetic sequences?

a)  \$latex 2, 8, 14, 20, …\$
$t_2-t_1=8-2 =6$
$t_3-t_2=14-8 =6$
$t_4-t_3=20-14 =6$

As there is a common difference of 6, the sequence is arithmetic.

b) \$latex -27, -21, -14, -7, …\$
$t_2-t_1=-21-(-27) =6$
$t_3-t_2=-14-(-21) =7$

As there is not a common difference, the sequence is not arithmetic.

c)  \$latex 0.37, 0.35, 0.33, 0.31, …\$
$t_2-t_1=0.35-0.37 =-0.02$
$t_3-t_2=0.33-0.35= -0.02$
$t_4-t_3=0.31-0.33=-0.02$

As there is a common difference of -0.02, the sequence is arithmetic.