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

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What is a quadratic sequence?

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

What is the 𝑛^th term of a quadratic sequence?

The 𝑛^th term of a quadratic sequence takes the form of: 𝑎𝑛² + 𝑏𝑛 + 𝑐.

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

𝑎 is the 2nd difference divided by 2.

𝑐 is the zeroth term.

How do you find the 𝑛^th 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 2^nd difference as 4.

Remember, 𝑛^th term = 𝑎𝑛² + 𝑏𝑛 + 𝑐

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

In this example, 𝑎 = 2 and 𝑐 = 1

We now have: 𝑛^th 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 𝑛^th term = 2𝑛² + 𝑏𝑛 + 1

The 4th term in the sequence is 33. So, substituting that into the formula for the 𝑛^th 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 𝑛^th 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 𝑛^th term of a quadratic sequence

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

Finding the 𝑛^th Term of a Quadratic Sequence Lesson Pack

Finding the 𝑛^th Term of a Quadratic Sequence Worksheet

Finding the 𝑛^th Term of a Quadratic Sequence Escape the Room Challenge Card

Finding the 𝑛^th 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!