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

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:

1^{st} term = a

2^{nd} term = a + d

3^{rd} term = a + 2d

4^{th} term = a + 3d

5^{th} 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:

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