# Arithmetic sequence

An arithmetic sequence is a number pattern in which there's a common difference between successive terms.

It's also known as an arithmetic progression.

## Example of an arithmetic sequence

Consider the following number pattern

 1 5 9 13 17

Look closely and you'll see that the common difference is +4

 1 +4 5 +4 9 +4 13 +4 17

You can find the nth term of an arithmetic progression using the formula

where d = common difference.

Try testing the formula on the 3rd term of the above sequence

1st term = 1

d(n - 1) = 4(3 - 1) = 4 x (3 - 1) = 4 x 2 = 8

3rd term  =  1  +  8  =  9

The formula seems to work but try it out on the 5th term to be sure!

1st term = 1

d(n - 1) = 4(5 - 1) = 4 x (5 - 1) = 4 x 4 = 16

5th term  =  1  +  16  =  17

## The common difference can also be negative

For instance, reversing the above progression gives

 17 13 9 5 1

common difference -4

 17 -4 13 -4 9 -4 5 -4 1

Let's work out the 4th term to check the nth term formula works for this reverse sequence

nth term = 1st term + d(n - 1)

1st term = 17

d(n - 1) = -4(4 - 1) = -4 x (4 - 1) = -4 x 3 = -12

4th term  =  17  +  -12  =  5

## A couple more examples

 -5 0 5 10 15

The common difference between successive terms in this progression is +5

 -5 +5 0 +5 5 +5 10 +5 15

Now say you want to find the 20th term. You could continue to write down the sequence until you reached this term

-5, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90

but you know the formula for the nth term works so why not use it?

nth term = 1st term + d(n - 1)

1st term = -5

d(n - 1) = 5(20 - 1) = 5 x (20 - 1) = 5 x 19 = 95

20th term  =  -5  +  95  =  90

## Try calculating the 10th term in this number pattern

 8 5 2 -1 -4

The common difference is -3

 8 -3 5 -3 2 -3 -1 -3 -4

Your working should look something like this

nth term = 1st term + d(n - 1)

1st term = 8

d(n - 1) = -3(10 - 1) = -3 x (10 - 1) = -3 x 9 = -27

10th term  =  8  +  -27  =  -19

## Summary of arithmetic sequences

If you know the 1st term in an arithmetic progression and the common difference (d) between successive terms, you can easily work out the nth term using the arithmetic sequence formula nth term = 1st term + d(n - 1)