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What is the formula for the nth term of an arithmetic sequence? (linear number sequences)

Updated on September 6, 2011

An arithmetic sequence is a sequence that increases or decreases by the same amount each time. Now if you call the first term in the sequence a and the difference between consecutive terms d then the:

1st term = a

2nd term = a + d

3rd term = a + 2d

4th term = a + 3d

5th term = a + 4d

And so on...

Therefore the nth term formula of an arithmetic number sequence is:

nth term = a + (n-1)d

So all you need to find the nth term is the first number in the sequence and the difference between consecutive numbers.

Example

Work out the nth term of this number sequence:

8,11,14,17,20...

The first term of this arithmetic number sequence is 8 and the sequence is going up by 3 from term to term.

Therefore, sub a = 8 and d = 3 into nth term = a + (n-1)d

nth term = 8 + (n-1)3

All you need to do now is multiply out the bracket and simplify:

nth term = 8 + 3n -3

nth term = 3n + 5

Example 2

Work out the nth term of this number sequence:

7,16,25,34,43...

The first term of this arithmetic number sequence is 7 and the sequence is going up by 9 from term to term.

Therefore, sub a = 7 and d = 9 into nth term = a + (n-1)d

nth term = 7 + (n-1)9

All you need to do now is multiply out the bracket and simplify:

nth term = 7 + 9n -9

nth term = 9n – 2

Example 3

Work out the nth term of this number sequence:

7,7.5,8,8.5,9...

The first term of this arithmetic number sequence is 7 and the sequence is going up by 0.5 from term to term.

Therefore, sub a = 7 and d = 0.5 into nth term = a + (n-1)d

nth term = 7 + (n-1)0.5

All you need to do now is multiply out the bracket and simplify:

nth term = 7 + 0.5n – 0.5

nth term = 0.5n + 6.5

The same formula can be applied for sequences that decrease by the same amount every time. All you need to remember is that the difference will be negative.

Example 4

9,5,1,-3,-7...

The first term of this arithmetic number sequence is 9 and the sequence is going down by 4 from term to term.

Therefore, sub a = 9 and d = -4 into nth term = a + (n-1)d

nth term = 9 + (n-1)-4

All you need to do now is multiply out the bracket and simplify:

nth term = 9 - 4n + 4

nth term = 13 – 4n

If you want an alternative method for finding the nth term of an arithmetic sequence then check out this article:

How to find the nth term of an increasing linear sequence

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