how to find the nth term

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Following on from the last blog on identifying different types of sequences, in this blog I will show you how to find the nth term of a linear sequence. This is a relatively simple process, but is incredibly useful.

What is the nth term, and why is it useful?

First of all, let me explain what the nth term of a sequence is. The nth term is a formula in terms of n that will find any term in the sequence that you want.

So let's say a sequence has nth term 4n + 1. If I wanted to find the 1^st term in the sequence, I can do that using the nth term. All I need to do is plug in n = 1.

n = 1 4×1 + 1 = 5

Therefore I know the first term in the sequence is 5.

I can find the 2^nd, 3^rd and 4^th terms as well too..

n = 2                      4×2 + 1 = 9
n = 3                      4×3 + 1 = 13
n = 4                      4×4 + 1 = 17So we know the sequence starts 5, 9, 13, 17.

You can clearly see here that the sequence we have generated is linear and goes up by 4 each time. So it is fairly easy to find the next few terms in the sequence.

But where the nth term becomes really useful is if I asked you to find the 200^th term in the sequence. If you had never heard of the nth term before, the only way you would know how to do this is to continue writing out the sequence until you get to the 200^th term… which would be time-consuming to say the least!

With the nth term, you can get it instantly. All you need to do is plug in n = 200.

n = 200 4×200 + 1 = 801So in a matter of seconds we know that the 200^th term in the sequence is 801. Much quicker than writing out 200 terms I'm sure you would agree!

There is another use of the nth term. Let's say we are still looking at the same sequence – 5, 9, 13, 17, …

Let's say I was asked "Is 665 in the sequence?".

Again, without the nth term, the only way to do this would be by writing out the sequence until we get to 665 and see whether it appears… but that's going to take even longer than the last question!

Again, the nth term can make this a lot easier and quicker for us.

All we need to do is take our nth term formula, and make it equal to the number we are testing (in this case it's 665). And then solve it! This will basically do the opposite of what we did in the previous example – it will tell us what number in the sequence 665 is… if the number we get is a whole number, that will make sense. However, if we don't get a whole number, this makes no sense. It makes sense to say that 665 is the 100^th term in a sequence, but not that it is the 100.6^th term in the sequence. The number must be an integer.

So in summary:

If the number you get is an integer, then this means the number is in the sequence.
If the number you get is not an integer, then this means the number is not in the sequence.

Let's see if 665 is in the sequence…

n is a whole number, therefore 665 is in the sequence! It is the 166^th number in the sequence.

Let's see if 287 is in the sequence.

How to find the nth term of a linear sequence

Now you know why it's useful, let's go through how to find the nth term of a linear sequence.

Example 1
Find the nth term of the sequence 7, 9, 11, 13, …

The first thing we need to do is find the common difference (in other words, what do you add or subtract the previous term by to get the next term).

In this linear sequence, the common difference is 2

In this case, the common difference is 2.

The common difference is always the number before the n.

So we know our nth term formula begins with 2n.

There is an extra bit though. To find this, we look at our sequence and go back a step – if there were one, what would the previous number in the sequence be?

Below, in green, I have worked out what the previous number would be. One way of thinking of it is "what number do you get 7 when you add 2?". It would be 5.

The previous term in this sequence would be 5

So this means we have to add 5 to our nth term formula.

This leaves us with 2n + 5.

We can check our answer to see if it is right.

We know the first four numbers in our sequence are 7, 9, 11 and 13.

So if we plug in n = 1, 2, 3 and 4, we should get those numbers. If we don't, something has gone wrong…

n = 1                      2×1 + 5 = 7 this matches our sequence
n = 2                      2×2 + 5 = 9 this also matches
n = 3                      2×3 + 5 = 11 this also matches
n = 4                      2×4 + 5 = 13 4 out of 4!

So there it is. All we need to do is find the common difference and the previous term. And we're done.

Example 2
Find the nth term of the sequence 5, 11, 17, 23, …

First, find the common difference…

In this linear sequence, the common difference is 6

So our formula starts with 6n.

Now find the previous term…

The previous term in this sequence would be -1

Therefore our nth term formula is 6n – 1.

Let's check the first three terms just to be sure…

n = 1                      6×1 – 1 = 5 this matches our sequence
n = 2                      6×2 – 1 = 11 this also matches
n = 3                      6×3 – 1 = 17 this also matches

Example 3
Find the nth term of the sequence 11, 8, 5, 2, …

First, find the common difference…

The common difference of this linear sequence is -3

In this example, the numbers are decreasing so our common difference is negative.

Our formula starts with -3n.

Now find the previous term…

The previous term of this linear sequence would be 14

Our nth term formula is -3n + 14.

Let's check the first three terms just to be sure…

n = 1                      -3×1 + 14 = 11 this matches our sequence
n = 2                      -3×2 + 14 = 8 this also matches
n = 3                      -3×3 + 14 = 5 this also matches

So there is an introduction to the nth term and how to find the nth term of a linear sequence. If you are studying the higher tier, you can read my guide on how to find the nth term of a quadratic sequence.

Below are 10 linear sequences. Have a go at finding the nth term of each of these sequences and put your answers in the comments or email them to sam@metatutor.co.uk and I'll let you know if you got them right. If you need someone to explain this method to you in person, book in a free taster session here.

1. 6, 10, 14, 18, …
2. 4, 9, 14, 19, …
3. 3, 10, 17, 24, …
4. 9, 13, 17, 21, …
5. 15, 11, 7, 3, …
6. 8, 17, 26, 35, …
7. 16, 19, 22, 25, …
8. -5, -2, 1, 4, …
9. 10, 2, -6, -14, …
10. -13, -17, -21, -25, …

For even more linear nth term questions, try our nth term worksheet.