In this section weâ€™re going to be looking at quadratic sequences. We shall explore how to find the nth term of a quadratic sequence. Here is an example of a sequence;

The differences between these terms in the sequences are not equal. This must mean that the sequence is not linear as it does not increase in equal steps as shown below;

We can see that the difference of this sequence changes at each step as it increases. Suppose we found the second difference;

The second difference of the sequence is constant. It does not change as the first difference. If the second difference is constant the sequence is referred to as a quadratic sequence, and therefore contains a n2 term.
Here is the sequence again and the terms labelled;

In the sequence above we can see that the first term is a 4, second term is a 7, third term is 19, and the fifth term is 28. Letâ€™s find n2 below;

If you look at the sequence you will realise that the sequence is always 3 more than n2 term as shown below;

This must mean that the rule to find any nth term in this sequence is;

We can prove this;

The above shows that our rule to find the nth term in the above sequence is valid.

## Example 2

Here is another quadratic sequence example;

First we find the first difference as we did above;

Then we find the second difference;

We can see that the second difference has a constant increment of 4. This must mean that itâ€™s a quadratic sequence and therefore as an n2
Now itâ€™s find the n2 next and compare;

Here we can see that when we compare the n2 sequence with the original sequence the outcome is not constant, we have to change n2 to make sure that it does. Letâ€™s try 2n2;

Now we have found a link by using 2n2. We can see that the sequence is always 5 more than the 2n2 sequence. Therefore the formula to find any nth term in this sequence is;

In the last example there was a twist. And you must familiarise yourself with this. In the first sequence we saw that the second difference was a 2 therefore we use n2. In the last example the second difference was 4 we found that 2n2 gave us a link to use with the sequence. You must use this trial and improvement method when this occurs to make sure that the link is always constant. If you continue to practice you will realise that;

• When the second difference is 2 the sequence starts with n2
• When the second difference is 4, the sequence starts with 2n2
• When the second difference is 8, the sequence starts with 4n2

And so on…

## 15 thoughts on “Quadratic Sequences”

1. john

when i first saw this i thought that i was never going to understand it since im hopeless at maths.
but really , this site is great and i really do understand quadratic equations now !!

2. Lucie Mcroosy

Hey,
Thanks for the article! Very helpful!
I was wondering what level this work is? And is there another quadratic sequence where you have to find a b and c?
Thanks,
Lucie Mcroosy

1. Author Post author

Hello Lucie,

This level is what 18-19 would probably do. It is college level maths or equivalent to A level.

I didn’t quite understand your last question.

1. Jason

This is for 18-19 years old students
wow,-i’ve fairly recently started year 8 and we’re learning this- i came here for help with my homework i had to find the nth term of two quadratic sequences
1))0,5,12,21,32-Which i think the answer is n2+2n-3
2)0,3,10,19,36
I’m having a lot of trouble with Q.2

1. Author Post author

Hello Jason,

Yes the first one is correct:
$n^2 + 2n -3$
Are you sure you have written the second sequence correctly?

May be the sequence is:
$0, 3, 10, 21, 36$
So then you will get;
$2n^2 - 3n + 1$
I thought I add the answer for you as well since you have spent a lot of time on it.

3. Jon

This is only level 7/8 maths at best. That is equivalent to C/B grade at GCSE. However, you would need to have something that includes a more complicated equation so that you could find “b” as well as “c” for the formula. I’ve just taught this to my year 8′s (12 year olds) but they are top set.

4. Author Post author

Your answer is better. I wasn’t sure this was in GCSE and 12 years old exams.

5. Lucie Mcroosy

Hey there Mr Jon
Well, I am 12 and second to top set. Top set does level 8, second does level 7. There are 6 sets altogether. We’ve learnt about the a b c. I think ‘a’ its a half the answer from the differences of the differences. ‘C’ is find the 0 term number. And I think ‘b’ is the hardest one-i think you have to do some simplifying and replacing the N’s with term number 1?
Am I wrong? I’ve kinda forgotten.
Lucie Mcroocy

6. Anonymous

Hi, What is the second difference is a constant but it is -5. for example,

Terms 4, -1, -11, -26, -46
Difference -5, -10, -15, -20
2nd Diff -5, -5, -5

How would N to the power of 2 come into this? Help. thanks.

1. Author

Hi,

There are no examples of how to handle such a situation.

The nth term is: Â½(-5nÂ² + 5n + 8)

7. Kaitlin

I really don’t understand why the n^ thing changes depending on the number? i’ve got online maths homework to do and i have to get over 80% and im just so confused:(((( This helped me a bit but im still so confused