This article explores quadratic sequences. We shall explore how to find the nth term of a quadratic sequence and workout the terms in a sequence given the nth term. Baisc knowledge of sequences will be useful here.

## What is a Quadratic sequence?

Sequence: A sequence is an ordered list of terms with a pattern

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 sequence

While the following sequence of numbers is known as a Quadratic sequence

Whenever the second difference is constant in a sequence, the sequence is said to be 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 n2s 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 3n2

In fact the following rules apply when determining the number of n2s

• Write out the original sequence above the terms of your number of n2
• Subtract the n2s 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 n2s to the formula for the residue and this will be the formula for the original sequence.

## Basic examples

Example:

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

nth term = n2 + 3
Explanation:

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. Let’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:

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

Explanation:

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 n2

Now it’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. Let’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… Therefore you always halve the second difference to get the number of n2s 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 2n2

## Harder examples (linear residues)

In a sequence the difference between the terms and the number of n2s 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 n2s which becomes the nth term of the quadratic sequence.

Example: Find the nth term for the following sequence;
1,1,3,7,13
nth term = n² – 3n + 3
Explanation:

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 n2 is involved since…

2/2 = 1

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;

1, 1, 3, 7, 13

…the nth term of the sequence is;

nth term = n² – 3n + 3
Example: Find the nth term for the following sequence; 5, 21, 47, 83, 128
nth term = 5n² + n – 1
Explanation:

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;

10/2 = 5

Therefore the nth term starts with 5n²…

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

5 – 5(1) = 0
21 – 5(2²) = 1
47 – 5(3²) = 2
83 – 5(4²) = 3

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

Therefore;

nth term = 5n2 + n -1

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