# GCSE Maths Revision: Finding the nth Term of a Quadratic Sequence

This blog explores the bare bones of quadratic sequences. If you’re looking for hands-on revision, check out the FREE revision bundle below.

## What is a quadratic sequence?

A quadratic sequence is one whose first difference varies but whose second difference is constant.

## What is the 𝑛th term of a quadratic sequence?

The 𝑛th term of a quadratic sequence takes the form of: 𝑎𝑛2 + 𝑏𝑛 + 𝑐.

We see why it’s called a quadratic sequence; the 𝑛th term has an 𝑛2 in it.

𝑎 is the 2nd difference divided by 2.

𝑐 is the zeroth term.

## How do you find the 𝑛th term of a quadratic sequence?

Look at the sequence: 3, 9, 19, 33, 51, …

The second difference is 4.

The zeroth term is the term which would go before the first term if we followed the pattern back.

Working backwards, we know the second difference will be 4.

So the first difference between the terms in position 0 and 1 will be 6 − 4 = 2.

And the zeroth term will be 3 − 2 = 1.

In the sequence: 3, 9, 19, 33, 51, …

we calculated the zeroth term as 1 and the 2nd difference as 4.

Remember, 𝑛th term = 𝑎𝑛2 + 𝑏𝑛 + 𝑐

where 𝑎 is the 2nd difference ÷ 2  and 𝑐 is the zeroth term

In this example, 𝑎 = 2 and 𝑐 = 1

We now have: 𝑛th term = 2𝑛2 + 𝑏𝑛 + 1

We need to find the value of 𝑏.

So far… in the sequence: 3, 9, 19, 33, 51, …

we know that the 𝑛th term = 2𝑛2 + 𝑏𝑛 + 1

The 4th term in the sequence is 33. So, substituting that into the formula for the 𝑛th term will help us to find the value of 𝑏:

2 × 42 + 4 × 𝑏 + 1 = 33

32 + 4𝑏 + 1 = 33

4𝑏 = 0 and 𝑏 = 0

Now that we have found the value of 𝑏, we know the 𝑛th term = 2𝑛2 + 1

Try checking it by working out, for example, the 3rd term and checking it with the sequence.

## Secondary resources for finding the 𝑛th term of a quadratic sequence

These KS3 maths resources are great for practising and applying finding the 𝑛th term of a quadratic sequence:

Finding the 𝑛th Term of a Quadratic Sequence Lesson Pack

Finding the 𝑛th Term of a Quadratic Sequence Worksheet

Finding the 𝑛th Term of a Quadratic Sequence Escape the Room Challenge Card

Finding the 𝑛th Term of a Quadratic Sequence and Problem Solving Lesson Pack

Fancy some more maths capers? Why not read our latest post on the iterative method for solving nonlinear equations below!

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