Quadratic sequences: finding the nth term step by step

What is a quadratic sequence?

A quadratic sequence is one where the first differences change by the same amount each time. The simplest is the square numbers: 1, 4, 9, 16, 25.

Sequence	1	4	9	16	25
First differences		3	5	7	9
Second differences			2	2	2

The first differences (3, 5, 7, 9) aren't the same, so it isn't a linear sequence. But the differences between them, the second differences, are all 2. A constant second difference is the test for a quadratic sequence.

Compare a linear sequence like 5, 8, 11, 14, which goes up by 3 every time. Its nth term is 3n + 2, as the nth term guide shows. A quadratic sequence has an nth term with n² in it, such as n² + 1 or 2n² − n + 4.

How to find the nth term of a quadratic sequence

To find the nth term of a quadratic sequence, work out the n² part from the second difference, take it away, and find the nth term of the linear sequence that's left.

1. Find the first differences, then the second differences.
2. Halve the second difference. That's the number in front of n². A second difference of 2 gives n², a second difference of 4 gives 2n², and a second difference of 6 gives 3n².
3. Write that n² sequence under the terms and take it away from each term.
4. Find the nth term of what's left. It's a linear sequence (or a single number repeated), so use the linear method.
5. Put the two parts together and check with a term you didn't use.

The finished quadratic nth term formula looks like an² + bn + c, where a, b and c are numbers. Step 2 finds a. Step 4 finds bn + c.

Worked example: 3, 9, 17, 27, 39

The first differences are 6, 8, 10, 12 and the second differences are all 2. Half of 2 is 1, so the nth term starts with n².

n	1	2	3	4	5
Sequence	3	9	17	27	39
n2	1	4	9	16	25
Sequence − n2	2	5	8	11	14

What's left, 2, 5, 8, 11, 14, goes up by 3. It's the 3 times table take away 1, so its nth term is 3n − 1. Put the parts together: the nth term is n² + 3n − 1.

Check with n = 5: 5² = 25, 3 × 5 = 15, and 25 + 15 − 1 = 39. That's the 5th term.

Worked example: 5, 10, 19, 32, 49

The first differences are 5, 9, 13, 17 and the second differences are all 4. Half of 4 is 2, so the nth term starts with 2n².

n	1	2	3	4	5
Sequence	5	10	19	32	49
2n2	2	8	18	32	50
Sequence − 2n2	3	2	1	0	−1

What's left goes down by 1 each time, so the difference is −1 and it starts with −n. The −1 times table goes −1, −2, −3, and you add 4 to get 3, 2, 1. So the linear part is −n + 4, and the nth term is 2n² − n + 4.

Check with n = 3: 3² = 9, so 2n² is 18. Then 18 − 3 + 4 = 19. That's the 3rd term.

When the second difference is negative

If the first differences get smaller, the second difference is negative and so is the number in front of n². For 9, 16, 21, 24, 25 the first differences are 7, 5, 3, 1 and the second differences are −2, so the nth term starts with −n². Taking away −n² is the same as adding n²: 9 + 1 = 10, 16 + 4 = 20 and 21 + 9 = 30. What's left, 10, 20, 30, is 10n. The nth term is 10n − n².

Check with n = 4: 10 × 4 = 40, 4² = 16, and 40 − 16 = 24.

Finding terms from a quadratic nth term

To find a term from a quadratic nth term, substitute the position for n and square first, before you multiply. For n² + 3n − 1, the 10th term is 100 + 30 − 1, which is 129, because 10² = 100.

For 2n², square n and then double. When n = 3, that's 2 × 9 = 18, not 6² = 36. This order comes from the usual order of operations, and it's the most common slip in this topic. Substituting into expressions is covered in algebra basics.

You can also ask whether a number is in the sequence. Is 50 a term of n² + 1? Then n² + 1 = 50, so n² = 49 and n = 7. It's the 7th term. Is 60 a term? Then n² = 59, which isn't a square number, so no. Students go on to solve harder versions of this with solving quadratic equations.

Common mistakes with quadratic sequences

• Using the second difference instead of half of it. A second difference of 4 means 2n², not 4n².
• Stopping at the first differences. If they aren't all the same, go one step further before deciding it isn't a pattern.
• Subtracting the wrong way round. Take the n² part away from the sequence, not the sequence from the n² part.
• Getting 2n² wrong. Square first, then multiply: when n = 4, 2n² is 32, not 64.
• Forgetting to check. Put a position you didn't use into the finished nth term and see that it gives the right term.

How to help at home

• Know the square numbers. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 come up in every question. They sit on the diagonal of the times table grid.
• Check the linear method first. The last step of every quadratic question is a linear nth term, so make sure the nth term of sequences like 2, 5, 8, 11 feels easy.
• Make one up together. Pick a rule like n² + 2n, write the first five terms, and ask your child to find the rule back.
• Use patterns. Square patterns of dots, or a staircase of blocks, grow in a quadratic way. Counting them helps the second difference make sense.
• Ask the school what's coming. Schools choose the order of GCSE topics across Years 10 and 11, so their scheme of work shows when sequences come up.

Where quadratic sequences fit

Key stage 3, from Year 7 to Year 9, covers linear sequences and their nth term. At key stage 4, in Years 10 and 11, every GCSE student meets quadratic sequences and square number sequences and should recognise them. Finding the nth term of a quadratic sequence is higher tier content: the DfE's GCSE subject content puts it in bold, for the more highly attaining students. See GCSE maths foundation and GCSE maths higher for the other topics in these years.

Quadratic nth term questions to try

1. Show that 4, 7, 12, 19, 28 is a quadratic sequence, then find its nth term.
2. Find the nth term of 2, 6, 12, 20, 30.
3. Find the nth term of 3, 10, 21, 36, 55.
4. Find the nth term of 1, 2, 5, 10, 17.
5. Write the first four terms of the sequence with nth term 3n² − 2.
6. Find the nth term of 11, 16, 19, 20, 19.

Answers

1. The first differences are 3, 5, 7, 9 and the second differences are all 2. The nth term is n² + 3.
2. n² + n (take away n² to leave 1, 2, 3, 4, 5)
3. 2n² + n (the second difference is 4, and taking away 2n² leaves 1, 2, 3, 4, 5)
4. n² − 2n + 2 (taking away n² leaves 0, −2, −4, −6, −8)
5. 1, 10, 25, 46
6. 8n + 4 − n² (the second difference is −2, and adding n² leaves 12, 20, 28, 36, 44)