How to find the nth term of a quadratic sequence

Following on from our previous blogs on identifying different types of sequences and finding the nth term of a linear sequence, in this blog I will show you how to find the nth term of a quadratic sequence. It is important to note that this topic is only examinable on the higher tier GCSE maths exam.

Let’s work through an example.

Example 1
Find the nth term of the quadratic sequence 3, 4, 7, 12, …

Before we start, because this is a quadratic sequence, we know our nth term formula is going to be of the form an² + bn + c. We just have to find a, b and c.

First, let’s find a. To find a, we find the difference of the differences in our sequence, and then divide this by 2.

The differences of the differences are in green below:

This gives us 2.

Now, divide this number by 2:

2 ÷ 2 = 1

We have found a (in other words, the coefficient of n² in our nth term formula).

Now, because we know our nth term contains 1n² (or just n²), we are going to write out our sequence, compare it to the sequence n² and then find the difference between these two. This difference will produce a sequence which we can use to find the rest of our nth term formula.

Below, in black, is our sequence.
Then, in red, is the sequence n².
Then, in green, is the difference between our sequence and the sequence n².

The differences between our sequence and the sequence n² now forms a linear sequence (in green above). This sequence should always be linear – if it isn’t, you have done something wrong. What we now need to do is find the nth term of this green sequence. We will need to add this on to n² – this will tell us our b and c. If you need a reminder of how to find the nth term of a linear sequence, you can re-read the previous blog.

The sequence has a difference of -2, and if there were a previous term it would be 4.

So the nth term of the green sequence is -2n + 4.

Adding this on to what we already knew, this means our nth term formula is n² – 2n + 4.

What I would strongly recommend at this stage is that you check your answer. Going back to why the nth term formula is useful, remember that the formula tells you any term in the sequence. We know from the question that the first term in the sequence is 3. So, if we plug 1 into the formula we should get 3. Likewise, we know that the second term in the sequence is 4, so if we plug 2 into the formula we should get 4. This allows us to check the formula we calculated is correct. And if we plug in 3, we should get 7.

n = 1                      1² – 2×1 + 4 = 1 – 2 + 4 = 3                              this matches our sequence!
n = 2                      2² – 2×2 + 4 = 4 – 4 + 4 = 4                              this also matches!
n = 3                      3² – 2×3 + 4 = 9 – 6 + 4 = 7                              this also matches!

Let’s do the fourth term as well, we know this should be 12…

n = 4                      4² – 2×4 + 4 = 16 – 8 + 4 = 12                         4 out of 4!

So we are sure our answer is correct. It’s always a nice feeling, not just in maths, when you give an answer and you know it is correct. I would recommend always try at least 2 terms, because you could always fluke one!

Example 2
Find the nth term of the quadratic sequence 1, 3, 9, 19, …

First, find a – the difference of the differences divided by 2.

The difference of the differences is 4 this time, so 4 ÷ 2 = 2, giving us a = 2.

So we know our sequence starts with 2n².

Now, compare our sequence with the sequence 2n² (this is just the sequence for n² but each term multiplied by 2).

This generates the linear sequence in green. We now need to find the nth term of this sequence.

The sequence has a difference of -4 and if there were a previous term it would be 3.

So the nth term of the green sequence is -4n + 3.

So our final nth term formula will be 2n² – 4n + 3.

Now, let’s check the first three terms…

n = 1                      2×1² – 4×1 + 3 = 2 – 4 + 3 = 1                         this matches our sequence
n = 2                      2×2² – 4×2 + 3 = 8 – 8 + 3 = 3                         this also matches
n = 3                      2×3² – 4×3 + 3 = 18 – 12 + 3 = 9                    this also matches

So we are confident our answer is correct.

Example 3
Find the nth term of the quadratic sequence 2, 3, 10, 23, …

First, find a.

6 ÷ 2 = 3.

So the nth term begins with 3n². Now compare our sequence to this.

Now find the nth term of the green sequence.

The sequence has a difference of -8 and if there were a previous term it would be 7.

So the nth term is -8n + 7.

Giving our final answer as 3n² – 8n + 7.

Check the first three terms…

n = 1                      3×1² – 8×1 + 7 = 3 – 8 + 7 = 2                         this matches our sequence
n = 2                      3×2² – 8×2 + 7 = 12 – 16 + 7 = 3                    this also matches
n = 3                      3×3² – 8×3 + 7 = 27 – 24 + 7 = 10                  this also matches

Example 4
Find the nth term of the quadratic sequence 8, 13, 20, 29, …

First, find a.

2 ÷ 2 = 1.

So the sequence begins with n². Now compare our sequence to this.

Now find the nth term of the green sequence.

The sequence has a difference of 2 and if there were a previous term it would be 5.

So the nth term is 2n + 5.

Giving our final answer as n² + 2n + 5.

Check the first three terms…

n = 1                      1² + 2×1 + 5 = 1 + 2 + 5 = 8                            this matches our sequence
n = 2                      2² + 2×2 + 5 = 4 + 4 + 5 = 13                          this also matches
n = 3                      3² + 2×3 + 5 = 9 + 6 + 5 = 20                          this also matches

So there is the method for finding the nth term of a quadratic sequence. It’s not the easiest of methods, but with some practice you can pick it up fairly quickly.

Try finding the nth term of these 5 quadratic sequences. Put your answers in the comments or email your answers to sam@metatutor.co.uk and I’ll let you know if you go them right. If you need someone to explain this to you in person, book in a free taster session.

1. 1, 7, 15, 25, …
2. 7, 12, 21, 34, …
3. 5, 12, 25, 44, …
4. 0, 9, 22, 39, …
5. 10, 19, 34, 55, …