You probably saw questions ‘What is the next number in the sequence?’ quite early in studying numbers.

### Next number in a sequence example

2 4 6 8 10 .. what is the next number?

Let’s just ignore the people who can see the answer straight off, and look for a method.

Look for the GAPS between each number. That is always the best start. In this example, the gap between each number is 2 – To put it another way, the numbers are going UP by 2 each time.

Once we have spotted that, we can move on through the sequence, adding two onto the last number each time.

… 12 14 16

Where we can get clever though is trying to find a general term in this sequence. This is a step up in effort, for sure.

What we do here is give all the numbers in a sequence a ‘place in the sequence’ which we do, in true algebra fashion, by using a letter. Its normal to use n in sequences. Questions will usually ask ‘find the nth’ term?’

We say we the first term in the sequence is n=1, then n=2 for the second, and so on. The general term then uses n in a formula. Let’s see how its done

### Finding the nth term

The first thing is to see the gap, as we saw before. In the first example, the gap was 2

1 4 7 10 13

In this sequence the gap is 3

So we start our nth term formula by putting this gap number in front of the n. 2n for the first example, 3n for this example.

So – does ‘3n’ give the sequence we’ve been asked to investigate?

No, because that sequence is

3 6 9 12 15 ..

But we can compare the two sequences – something we do a lot when looking for nth term formulas – and see our sequence is 2 less for eeach term than 3n: 3 6 9 …

so we have our formula 3n – 2

We can check, say, the 5th term – 5 x 3 – 2 = 15 – 2 = 13. That matches what we were given. We can now confidently predict the 100th term

100 x 3 – 2 = 200 – 2 = 298

### Is the number in the sequence?

Another common question is – is 100 in the sequence 1 4 7 10….

This question has not asked you to find the nth term – but that is the route to finding the answer.

We have already found the nth term for this sequence. This means we need to find n where

3n – 2 = 100.

We start to solve this like an equation, by taking 2 from both sides

3n = 98.

Now we hit a hitch. n is not going to be a whole number because 3 is not a factor of 98. In sequences we are only interested in cases where n IS a whole number.

from this we can say that 100 is NOT in the sequence because there is no n where 3n – 2 = 100

### More Practice

For more practice, see this website