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Question

Find the n^th term for this sequence: 1, 4, 7, 10...

• Here n = position and s = term.
• Eg when n = 2 (2nd position), s = 4.

Answer

First find the common difference between each term and the next.

In this case, every time you move along one position (n + 1), the term goes up by three (s + 3). This is three times as much and this tells us part of our formula:

s = 3 x n

To find out if this is the full formula, we substitute in one of the terms:

Term 1

When we substitute n = 1 into this formula, we find that it doesn’t work:

s = 3 x 1

= 3

But the first term is 1 so this isn't the full equation.

To get from s = 3 to s = 1 all we have to do is subtract 2:

s = 3 x 1 - 2

= 3 - 2

= 1

Now try this for some other terms to make sure your rule works:

Term 2

s = 3 x 2 - 2

= 6 - 2

= 4 (Correct)

Term 4

s = 3 x 4 - 2

= 12 - 2

= 10 (Correct)

Now we have established our formula:

s = 3 x n - 2

or more simply:

s = 3n -2

This method will always work for sequences where the difference between terms stays the same (common difference).