# Quadratic Sequences

## What is a Quadratic sequence?

Each number in a sequence is known as a term. We identify a term by its position in the sequence. For example the first term is the term that occurs first in a sequence. The 5th term is the term that occurs in the fifth place of the sequence.

In an arithmetic (linear) sequence the difference between each term is constant while in a quadratic sequence the difference between each term is not constant The following sequence is known as an arithmetic sequenceWhile the following sequence of numbers is known as a Quadratic sequence

## Important rules

There a few basic rules you should follow when trying to find a formula that can be used to find the nth term in a sequence, i.e the 6th or 100th term in a quadratic sequence.

- Halve the second difference to get the number of n
^{2}s for example if you the second difference of a sequence is +6 (which would be constant) you would divide 6 by 2 to get 3 therefore this becomes 3n^{2}In fact the following rules apply when determining the number of n

^{2}s- If the second difference is 2, you start with n
^{2}. - If the second difference is 4, you start with 2n
^{2}. - If the second difference is 6, you start with 3n
^{2}.

- If the second difference is 2, you start with n
- Write out the original sequence above the terms of your number of n
^{2} - Subtract the n
^{2}s from the sequence to give the Residue - The residue will either be constant or a linear sequence. If it is a linear sequence then work out its formula.
- Finally add the number of n
^{2}s to the formula for the residue and this will be the formula for the original sequence.

## Basic examples

Find a general expression for the nth term of the following sequence;

^{2}+ 3

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 n^{2} 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. Let’s find n^{2} below;

If you look at the sequence you will realise that the sequence is always 3 more than n^{2} 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.

Find an expression for the nth term of the following sequence;

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 it’s a quadratic sequence and therefore as an n^{2}

Now it’s find the n^{2} next and compare;

Here we can see that when we compare the n^{2} sequence with the original sequence the outcome is not constant, we have to change n^{2} to make sure that it does. Let’s try 2n^{2};

Now we have found a link by using 2n^{2}. We can see that the sequence is always 5 more than the 2n^{2} 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 n^{2}. In the last example the second difference was 4 we found that 2n^{2} 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 n
^{2} - When the second difference is 4, the sequence starts with 2n
^{2} - When the second difference is 8, the sequence starts with 4n
^{2}

And so on… Therefore you always halve the second difference to get the number of n^{2}s for example if the second difference of a sequence is +4 (which would be constant) you would divide 4 by 2 to get 2 therefore this becomes 2n^{2}

## Harder examples (linear residues)

In a sequence the difference between the terms and the number of n^{2}s my not be constant but maybe linear. If the sequence is linear then we work out the nth term for the linear difference and add it to the n^{2}s which becomes the nth term of the quadratic sequence.

We take the same steps as we did above. We have to find the first and second difference.

Look at the second difference we can see that just 1 n^{2} is involved since…

We continue as we did before by finding the difference between n² and the sequence terms

The residue is not constant. We therefore have to find the nth term of the residue. Given that;

We simply add the residue nth term to n². So given the sequence;

…the nth term of the sequence is;

First find the first difference and then the second difference of the sequence;

Since the second difference is constant that must mean at the sequence is a quadratic sequence. Therefore the sequence involves n² We divide the second difference by two to determine the number of n² involved;

Therefore the nth term starts with 5n²…

We must now work out how the the sequence is from n²

Notice that the residue is not constant here. We therefore have to find the nth term of the residue.

Therefore;

^{2}+ n -1

You can prove it for your self by finding all the terms in the original sequence.

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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 !!

but what if there is more than one nth term

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?

Please help.

Thanks,

Lucie Mcroosy

Hello Lucie,

Glad you liked the article.

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.

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

PLEASE HELP!

sorry

just ignore my last comment ive read on

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.

Hello Author,

Thanks for your answer. Don’t worry about my last question, I was probably dreaming.

Thanks ever so much,

Lucie Mcroosy.

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.

Thanks for the reply Jon,

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

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

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.

Hi,

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

The nth term is: ½(-5n² + 5n + 8)

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

YOu posted that 2 years ago but for each term the number n changes so if you want the 100th term, n is 100

link to a good online calculator for this??

the question is

12 , 25 , 42 , 63 , 88

can u pls help me find the sequence ?

Hello Chasm,

You must show that you have tried so that we know where you’re stuck. Have you tried it? So you will need to find the first difference. You do that by subtracting the previous term from the next term. For example:

[ 25-12=13, 42-25=17, 63-42=21, 88-63=25 ]

So now you know that the first difference is:

[ 13, 17, 21, 25 ]

Using the examples above, I am sure you can continue from here.

Hi, what about the sequence : 3,10,21,36,55 ?

Hi,

You must show that you tried. Start by finding the first difference. Have you tried that?

Hi!

The example that you’ve shown is to find the nth term of nonlinear sequence with positive second difference. How about sequence with negative second difference? Like this………..

Terms: 36, 35, 32, 27, 20, 11

Difference: -1, -3, -5, -7, -9

2nd difference: -5, -5, -5, -5

How to find?

Hi,

I have this sequence: 2.7,15,26..

The second difference is 3 and I’m not sure what to do?!

I have worked on this equation for the past hour.

9,13,13,9,-11,-27,-47

I am in Grade 9 High Level IGCSE Maths, and I can’t do this equation. I know the second difference is -4, and so according to Un=an^2+bn+c, a should be -2. But I can’t figure out b or c; a+b is supposed to be 4 (the first difference) and a+b+c should be 9 (the first term) and so b SHOULD be 6 and c SHOULD be 5 but it simply doesn’t seem to match up to the equation. I really have tried this all, but it isn’t working. PLEASE HELP! Our teacher likes to challenge us, and I’ve done the rest of the homework but this one has got me completely stuck.

There is another way to solve this quadratic sequence faster and easier.

hi whats the nth term of -1, 3, 11, 23, 39? i have tried but cant work it out?

is 0,1,4,9 a quadratic sequence

Good stuff

There are two mistakes in the examples – the first difference in 1,1,3,7,13 – is 0,2,4,6 – and 5,21,47,83,128 – should read 5,21,47,83,129

hi. I have been looking for the answer several times but I couldn’t figure it out.

What is the nth term for 0,1,6,15 28.

Thanks

Hiya!

I have to find the nth term (the formula) for 84,60,40,24,12,4 and i can see that the second difference is 40 so it would be 2nsquared and then following the rest of your method i get 2nsquared – 30 but this doesn’t work? is it because it is decreasing not increasing can someone help please?

I was wondering as to how to find the sequence rule in terms of n, when the second difference is a multiplied constant pattern (rather than an added one)

for example

1, 4, 16, 64, … (first term=4, second term = 4, etc)

[First difference: +3, +12, +48. Therefore, the second difference can be: x4, x4, x4]

it should be something along the lines of n^4, i figured at first… but i can’t seem to figure it out.

Help would be appreciated, thanks.

I am a Mathematics teacher and this is a new meyhod! Amazing that there is always something to learn. But is there not a msistake with example: 1; 1; 3; 7; 13: The first difference iss 0; 2; 4; 6?

Hi that is a graphical rendering error. It will be correct.