Finding the nth term of a sequence (the easy kind)

Everyone dreads the nth term questions, but once you get the hang of them, they're not actually all that hard.

A typical question might be: A sequence of numbers starts 2, 5, 8, 11... . What is the next number in the sequence? Find an expression for the nth term of the sequence.

The method

The next number is 14, obviously - you just add three each time. The difference between the terms is 3 - a term is simply a number in the sequence.

To find the nth term of the sequence, we look for a different sequence that goes up in threes - a good one would be the three times table.

A sidetrack that explains things

The nth term of the three times table is 3n. What does that even mean? It means that if you pick a number - say, 74 - I can tell you that the 74th number in the three times table is three times 74, or 222.

In that case, you picked 74 to be n and I replaced the n in the equation with 74. If I wanted the 100th term, n would be 100.

But the three times table isn't the same as this sequence!

Quite right. The sequence goes 2, 5, 8, 11... and the three times table goes 3, 6, 9, 12.... To get from the three times table to the sequence, we have to take away one. So the expression for the nth term of that series is 3n - 1.

Can you give me another example?

Why, of course! Let's look at -4, 3, 10, 17.... Here, the difference between terms is 7, so we're going to start from the seven times table - which has the expression 7n.

To get from 7, 14, 21... to -4, 3, 10..., we need to take away 11 - so the expression for the nth term of the sequence -4, 3, 10... is 7n - 11.