Mathématiques

Quadratic Sequences and Second Differences

If the second differences are constant the sequence is quadratic, and the coefficient of n squared is half that second difference. Worked on 2, 5, 10, 17, 26.

Quadratic sequence

A sequence whose second differences are all the same non-zero number, which means its nth term rule contains an n² term.

Où les élèves le rencontrent: Grade 9 mathematics, and in the nth term questions on higher-tier GCSE papers and Extended IGCSE papers.

La réponse en bref
A quadratic sequence is one whose second differences are constant, which means its nth term rule contains an n² term. In 2, 5, 10, 17, 26 the first differences are 3, 5, 7, 9 and the second differences are all 2, so the rule is n² + 1.

Un exemple

2, 5, 10, 17, 26 → first differences 3, 5, 7, 9 → second difference 2 every time

The first differences are not constant, so the sequence is not arithmetic. The second differences are, and they are 2, so the coefficient of n² is 2 ÷ 2 = 1. Subtracting the values of n² — that is 1, 4, 9, 16, 25 — from the sequence leaves 1, 1, 1, 1, 1, so the rule is n² + 1.

The two-row test

Write the sequence in a row, and underneath it the differences between neighbouring terms. If that row is constant the sequence is arithmetic and there is nothing quadratic about it. If it is not constant, take differences again to make a third row.

A constant third row is the signature of a quadratic sequence. It is not a coincidence: squaring produces first differences that grow by a fixed amount, so differencing twice flattens them out. Sequences needing three rounds of differencing before settling are cubic, and are rare at this level.

Line each row up in the gaps rather than under the terms. Five terms give four first differences and three second differences, so a row that comes out the same length as the one above it has an extra number in it somewhere.

Reading the n² coefficient off the second difference

The coefficient of n² is always half the second difference. A second difference of 2 gives n², a second difference of 6 gives 3n², and a second difference of −4 gives −2n².

The reason is visible in the square numbers themselves. In 1, 4, 9, 16, 25 the first differences are 3, 5, 7, 9 and the second difference is 2, so plain n² carries a second difference of 2. Multiplying a rule by 3 multiplies all its differences by 3 as well, which is why 3n² arrives with a second difference of 6.

What the second difference does not tell you

It fixes the n² part and nothing else. Both n² + 1 and n² + 10n have a second difference of 2, and so does every rule of the form n² + bn + c. That is why the second difference is a first move rather than an answer.

The rest is recovered by subtracting the square numbers from the original sequence. Whatever remains is arithmetic or constant, and it is handled with the ordinary nth term rule for a linear sequence — which is why quadratic sequences are taught after arithmetic ones and not before.

Questions fréquentes

How is a quadratic sequence different from an arithmetic one?

An arithmetic sequence has constant first differences, so it climbs by the same amount every time and its rule is linear. A quadratic sequence has first differences that themselves change by a constant amount, so it accelerates. One round of differencing settles an arithmetic sequence; a quadratic one needs two.

Why is the coefficient of n² half the second difference?

Because the square numbers 1, 4, 9, 16 have first differences 3, 5, 7 and therefore a second difference of exactly 2. Every quadratic rule is a multiple of n² plus a linear part, and the linear part contributes nothing to the second differences, so the second difference is always twice the n² coefficient.

Can the second differences be negative?

Yes. A second difference of −2 gives a coefficient of −n², and the sequence eventually turns and falls, the way the height of a thrown ball does. The method is unchanged: halve the second difference, keeping the sign, and carry on.

Is 2, 4, 8, 16, 32 a quadratic sequence?

No. Its first differences are 2, 4, 8, 16 and its second differences are 2, 4, 8 — still not constant, and differencing further never settles. Dividing consecutive terms gives 2 every time, so it is geometric, and no polynomial rule in n will produce it.

Dernière mise à jour 21 août 2026