Quadratic sequences and the nth term

A quadratic sequence is a sequence whose second differences are all the same. Its nth term has the form an² + bn + c, where a is half the second difference. For example, 2, 9, 20, 35, 54 has first differences 7, 11, 15, 19 and a constant second difference of 4, so a = 2, and the quadratic nth term works out as 2n² + n − 1.

What is a quadratic sequence?

In a linear sequence such as 5, 8, 11, 14, the gap between terms is the same every time. In a quadratic sequence the gaps themselves change, but they change by the same amount each time. You find this by taking differences twice:

Constant first differences mean a linear sequence. Constant second differences mean a quadratic sequence. The square numbers 1, 4, 9, 16, 25 are the simplest quadratic sequence: their nth term is n², and their second difference is 2.

Here x is the position n and y is the term. The terms sit on a parabola, which is why the sequence is called quadratic. A linear sequence would sit on a straight line.

How to find the nth term of a quadratic sequence

1. Find the second difference. Halve it to get a, the number in front of n².
2. Write out an² for n = 1, 2, 3, 4, 5.
3. Subtract an² from each term of the sequence. What is left is a linear sequence (or a constant).
4. Find the nth term of what is left, bn + c, in the usual way.
5. Add the parts: nth term = an² + bn + c. Check it with one of the original terms.

Example 1: 3, 8, 15, 24, 35

First differences 5, 7, 9, 11; second difference 2, so a = 2 ÷ 2 = 1.

What is left, 2, 4, 6, 8, 10, is the linear sequence 2n. So the nth term is n² + 2n. Check: the 3rd term is 9 + 6 = 15.

Example 2: 2, 9, 20, 35, 54

The second difference is 4, so a = 2. Subtract 2n²:

0, 1, 2, 3, 4 goes up by 1 each time and starts one below n, so it is n − 1. The nth term is 2n² + n − 1. Check: the 4th term is 32 + 4 − 1 = 35.

Example 3: a negative second difference

Find the nth term of 10, 13, 12, 7, −2.

First differences 3, −1, −5, −9; second difference −4, so a = −2. Subtract −2n², which means adding 2n²:

12, 21, 30, 39, 48 goes up by 9, so it is 9n + 3. The nth term is −2n² + 9n + 3. Check: the 2nd term is −8 + 18 + 3 = 13.

Example 4: a half in front of n²

The triangular numbers 1, 3, 6, 10, 15 have a second difference of 1, so a = ½. Subtract ½n² (0.5, 2, 4.5, 8, 12.5) to leave 0.5, 1, 1.5, 2, 2.5, which is ½n. The nth term is ½n² + ½n, often written n(n + 1)/2.

You can generate any sequence from its nth term and see the differences in the Sequences and Series calculator.

Another method: three equations

If you write out the general sequence an² + bn + c, its first term is a + b + c, its first difference is 3a + b, and its second difference is 2a. Match these to your sequence. For 2, 9, 20, 35, 54:

So a = 2, then b = 7 − 6 = 1, then c = 2 − 2 − 1 = −1, giving 2n² + n − 1 again. This method is quicker once you have learned the three expressions, but the subtraction method shows more working.

Using the nth term

Example 5: a term far along

Find the 20th term of 2n² + n − 1. Substitute n = 20: 2 × 400 + 20 − 1 = 819.

Example 6: is a number in the sequence?

Is 224 a term of n² + 2n? Set the nth term equal to 224 and solve:

n = 14 is a positive whole number, so 224 is the 14th term. Try the same with 200: n² + 2n − 200 = 0 gives n ≈ 13.2, not a whole number, so 200 is not in the sequence.

Check your understanding.

Common mistakes

• Using the whole second difference as a. a is half of it: a second difference of 4 means 2n².
• Subtracting n² instead of an². Subtract the full an² term, with its coefficient.
• Sign errors with a negative a. Subtracting −2n² means adding 2n².
• Squaring the coefficient. 2n² means 2 × n², so for n = 3 it is 2 × 9 = 18, not 6² = 36.
• Not checking. Substitute n = 1 or n = 2 into your answer. It should give the first or second term.

Practice questions

1. Find the nth term of 2, 7, 14, 23, 34.
2. Find the nth term of 6, 15, 28, 45, 66.
3. Find the nth term of 4, 7, 8, 7, 4.
4. Is 120 a term of n² + 2n? If so, which one?

Answers: 1) n² + 2n − 1. 2) 2n² + 3n + 1. 3) −n² + 6n − 1. 4) Yes: n² + 2n = 120 gives n = 10, so it is the 10th term.

Practise with a short timed paper

Sequences come up in GCSE Maths Higher algebra. The daily GCSE Maths Higher paper is a short timed paper of 8 questions, with a new one every day.

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