Quadratic and Other Non-linear Sequences — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers sequences whose terms do not change by a constant amount. By the end of this guide you should be able to recognise a quadratic sequence from its differences and find its nth term.

You should also be able to work with geometric sequences, Fibonacci-type sequences, and the special sequences built from square, cube and triangular numbers, as well as use an nth term rule to find a distant term or test whether a number belongs to the sequence.

The organising idea is that the differences tell you what kind of sequence you have, and you keep taking differences until they settle. A constant first difference means the sequence is linear. Not constant? Take the differences of those differences. If the second differences are constant, the sequence is quadratic and contains an n² term. If the differences never settle but the ratios do, it is geometric. One habit — write the differences underneath — sorts every sequence into its type and hands you the first part of the rule at the same time.

Key terms and definitions

Sequence — an ordered list of numbers following a rule.

Term — one number in the sequence. The first term is at position n = 1.

First differences — the gaps between consecutive terms.

Second differences — the gaps between the first differences.

Quadratic sequence — one whose nth term contains n², identified by constant second differences.

Geometric sequence — one where each term is multiplied by a fixed common ratio.

Fibonacci-type sequence — one where each term is the sum of the two before it.

nth term rule — a formula giving the term at any position directly.

Core concepts

Identifying the type

Write the differences underneath the sequence before anything else.

For 2, 5, 10, 17, 26 the first differences are 3, 5, 7, 9 — not constant, so not linear. The second differences are 2, 2, 2 — constant, so the sequence is quadratic.

For 2, 6, 18, 54 the first differences are 4, 12, 36 and the second differences are 8, 24 — neither settles. But dividing each term by the one before gives 3, 3, 3, so the sequence is geometric.

For 1, 1, 2, 3, 5, 8 neither differences nor ratios settle, but each term is the sum of the two before it — a Fibonacci-type sequence.

That short check identifies every sequence on the specification.

Finding the nth term of a quadratic sequence

The coefficient of n² is half the second difference. That is the key fact, and the rest follows mechanically.

For 2, 5, 10, 17, 26 the second difference is 2, so the coefficient of n² is 1, and the rule contains n².

Now subtract n² from each term. The squares at positions 1 to 5 are 1, 4, 9, 16, 25, and subtracting gives 1, 1, 1, 1, 1.

What remains is an ordinary sequence you already know how to handle — here a constant 1 — so the nth term is n² + 1.

Check on a term other than the first: at n = 4, 16 + 1 = 17. ✓

When the remainder is linear

Often the subtraction leaves a linear sequence rather than a constant, and then the usual linear method finishes the job.

Take 4, 11, 22, 37, 56. The first differences are 7, 11, 15, 19 and the second differences are 4, so the coefficient of n² is 2.

Subtract 2n² from each term. At positions 1 to 5, 2n² gives 2, 8, 18, 32, 50, and subtracting leaves 2, 3, 4, 5, 6.

That remainder is linear with a common difference of 1, and at n = 1 it gives 2, so its rule is n + 1.

The full nth term is 2n² + n + 1.

Always verify on a later term: at n = 3, 2(9) + 3 + 1 = 22. ✓

Why halving the second difference works

It is worth knowing rather than taking on trust.

The sequence n² gives 1, 4, 9, 16, whose first differences are 3, 5, 7 and whose second differences are 2. So a pure n² sequence has a second difference of 2.

Multiplying the sequence by a doubles every difference, so a sequence containing an² has a second difference of 2a. Halving recovers a.

That also explains why the second difference is always constant for a quadratic: the squares grow by an amount that itself grows steadily.

Geometric sequences

Each term is the previous one multiplied by the common ratio, found by dividing any term by its predecessor.

A ratio above 1 makes the sequence grow; a ratio between 0 and 1 makes it shrink, as in 80, 40, 20, 10 with ratio ½; a negative ratio makes the signs alternate, as in 3, −6, 12, −24 with ratio −2.

The nth term is the first term multiplied by the ratio (n − 1) times. For a first term of 5 and a ratio of 2, the 6th term is 5 × 2⁵ = 160 — the power is one less than the position, because the first term has not been multiplied at all.

Fibonacci-type sequences

Each term is the sum of the two before it, so two starting terms define the whole sequence.

These have no simple nth term rule at GCSE. Questions ask you to continue the sequence, or to work backwards to find a missing earlier term.

Working backwards uses subtraction: if a sequence runs a, b, 7, 11, 18, then b = 11 − 7 = 4 and a = 7 − 4 = 3.

Some questions use algebra, giving the first two terms as expressions and asking for a later one in terms of them.

The special sequences

Recognising these saves time, since they appear inside other sequences constantly.

Square numbers: 1, 4, 9, 16, 25, with nth term n².

Cube numbers: 1, 8, 27, 64, with nth term n³.

Triangular numbers: 1, 3, 6, 10, 15, where the differences are 1, 2, 3, 4. Their nth term is n(n + 1)/2, and their second differences are constant at 1, so they are quadratic.

A sequence that is close to a familiar one is often that sequence shifted: 2, 5, 10, 17 is simply the square numbers plus 1.

Using the nth term rule

Once you have a rule, any term follows by substitution. For n² + 1, the 20th term is 400 + 1 = 401, with no listing required.

To test whether a number belongs, set the rule equal to it and solve. The number is in the sequence only if n comes out as a positive whole number.

Is 65 a term of n² + 1? Setting n² + 1 = 65 gives n² = 64, so n = 8. That is a positive whole number, so yes, and it is the 8th term.

Is 50 a term? Then n² = 49... which gives n = 7, so yes. But for 40, n² = 39, and the square root is not a whole number, so no.

Worked examples

Example 1: A quadratic nth term

Find the nth term of 3, 9, 19, 33, 51.

First differences: 6, 10, 14, 18. Second differences: 4, 4, 4 — constant, so the sequence is quadratic.

Half the second difference is 2, so the rule contains 2n².

At positions 1 to 5, 2n² gives 2, 8, 18, 32, 50. Subtracting from the original terms leaves 1, 1, 1, 1, 1.

The remainder is a constant 1, so the nth term is 2n² + 1.

Check at n = 4: 2(16) + 1 = 33. ✓

Example 2: A quadratic with a linear remainder

Find the nth term of 5, 12, 23, 38, 57.

First differences: 7, 11, 15, 19. Second differences: 4, so the coefficient of n² is 2.

Subtract 2n² — that is 2, 8, 18, 32, 50 — leaving 3, 4, 5, 6, 7.

That remainder is linear with a common difference of 1, and it starts at 3 when n = 1, so its rule is n + 2.

The nth term is 2n² + n + 2.

Check at n = 5: 2(25) + 5 + 2 = 57. ✓

Example 3: Working backwards in a Fibonacci-type sequence

A Fibonacci-type sequence runs p, q, 10, 16, 26. Find p and q.

Each term is the sum of the two before it, so 10 + q = 16, giving q = 6.

Then p + q = 10, so p = 10 − 6 = 4, and q = 6.

Check forwards: 4, 6, 10, 16, 26 — each term is indeed the sum of the two before it. ✓

Common mistakes and how to avoid them

Forgetting to halve the second difference. A second difference of 6 gives 3n², not 6n².

Stopping after the n² term. Subtract it from every term and deal with what remains.

Checking only the first term. Any rule can be made to fit the first term. Verify on a later one.

Assuming a non-linear sequence must be geometric. Take second differences before checking ratios.

Using the wrong power in a geometric sequence. The nth term uses the ratio to the power n − 1.

Answering a membership question with a decimal. The decimal is the working; the answer is that the number is not a term.

Muddling triangular numbers with a linear sequence. They are quadratic, with constant second differences of 1.

Exam technique for "Quadratic and Non-linear Sequences"

Write both rows of differences underneath the sequence as your first line. It identifies the type and gives the n² coefficient, and it usually earns a method mark on its own.

Set the subtraction out as a table — the original terms, the an² values, and the remainder — so each stage is visible.

Verify the finished rule on the last given term, not the first.

For a geometric sequence, state the common ratio explicitly before using it, and count the power carefully.

When asked whether a number is in a sequence, finish with a sentence rather than leaving a value of n.

If the second differences are not quite constant, re-check the subtraction before assuming the sequence is not quadratic — an arithmetic slip is far more likely.

Quick revision summary

Keep taking differences until they settle. Constant first differences mean linear; constant second differences mean quadratic; constant ratios mean geometric.

For a quadratic sequence, the coefficient of n² is half the second difference. Subtract that an² term from every term, and what remains is an ordinary constant or linear sequence.

So 2, 5, 10, 17 has second difference 2, giving n², and the remainder 1 — so the rule is n² + 1.

Verify on a later term, since any rule can be forced to fit the first.

In a geometric sequence, the ratio comes from dividing a term by the one before, and the nth term uses the ratio to the power n − 1. A ratio below 1 shrinks the sequence; a negative ratio alternates the signs.

Fibonacci-type sequences add the two previous terms, have no simple nth term, and are worked backwards by subtraction.

Recognise squares (n²), cubes (n³) and triangular numbers (n(n + 1)/2, which are quadratic).

To test membership, solve the rule for n — the number belongs only if n is a positive whole number.