Sequences and the nth Term — AQA GCSE Maths Revision Notes
What you'll learn
A sequence is an ordered list of numbers that follows a rule. By the end of this guide you should be able to recognise arithmetic and geometric sequences, continue them, find the nth term of an arithmetic sequence, and use that rule to find any term you are asked for.
You should also be able to test whether a given number belongs to a sequence, recognise the special sequences that appear repeatedly at GCSE — square, cube, triangular and Fibonacci-type — and handle quadratic sequences, where the differences themselves form an arithmetic sequence.
The organising idea is the difference. Looking at what happens between one term and the next tells you which kind of sequence you have and gives you the first part of its rule straight away. If the difference is constant, the sequence is arithmetic and that difference is the coefficient of n. If the difference is not constant but the ratio is, the sequence is geometric. If neither is constant, take the differences of the differences — and if those are constant, the sequence is quadratic. One habit, three outcomes.
Key terms and definitions
Sequence — an ordered list of numbers generated by a rule.
Term — one number in the sequence. The nth term is the one in position n, where the first term is n = 1.
Arithmetic sequence — one with a constant difference between consecutive terms.
Common difference (d) — the fixed amount added each time. It is negative for a decreasing sequence.
Geometric sequence — one where each term is multiplied by a fixed number.
Common ratio (r) — that fixed multiplier, found by dividing any term by the one before it.
nth term rule — a formula giving the term in position n directly, without working through all the earlier terms.
Position-to-term rule — another name for the nth term rule, and a reminder of what it does: you put the position in, and it gives the term out.
Core concepts
Recognising the type
Write the differences underneath the sequence before doing anything else.
For 3, 7, 11, 15, the differences are 4, 4, 4. Constant, so the sequence is arithmetic.
For 2, 6, 18, 54, the differences are 4, 12, 36 — not constant. But dividing each term by the one before gives 3, 3, 3, so the sequence is geometric with common ratio 3.
For 2, 5, 10, 17, the differences are 3, 5, 7. Not constant, and the ratios are not constant either. But the differences of those differences are 2, 2 — constant, so the sequence is quadratic.
The nth term of an arithmetic sequence
The coefficient of n is always the common difference. That is the whole first step.
For 3, 7, 11, 15, the difference is 4, so the rule begins 4n. Now check what 4n actually gives at n = 1: it gives 4, but the first term is 3. So the rule needs 1 subtracting: 4n − 1.
Test it on a later term to be sure. At n = 4, 4(4) − 1 = 15, which matches. ✓
The same method works for decreasing sequences, where the difference is negative. For 20, 17, 14, 11, the difference is −3, so the rule begins −3n. At n = 1 that gives −3, but the first term is 20, so add 23: −3n + 23, usually written 23 − 3n.
The adjustment is the commonest place to go wrong, and the fix is always the same: substitute n = 1 and compare with the actual first term.
Using the nth term
The point of having a rule is that it reaches any term in one step. For 4n − 1, the 50th term is 4(50) − 1 = 199, without listing anything.
The rule also answers the reverse question. To test whether a number is in the sequence, set the rule equal to it and solve. The number is in the sequence only if n comes out as a positive whole number, because positions cannot be fractions or negatives.
Is 100 a term of 4n + 1? Set 4n + 1 = 100, so 4n = 99 and n = 24.75. Not a whole number, so 100 is not in the sequence.
Geometric sequences
Each term is the previous one multiplied by the common ratio, found by dividing any term by its predecessor.
For 80, 40, 20, 10, dividing gives a ratio of ½, so the sequence decreases. A ratio between 0 and 1 always produces a decreasing sequence, even though multiplication is involved.
A negative ratio makes the terms alternate in sign: 3, −6, 12, −24 has ratio −2.
To find a later term, multiply repeatedly by the ratio. A geometric sequence with first term 5 and ratio 2 has 6th term 5 × 2⁵ = 160 — note the power is 5, not 6, because the first term has not been multiplied at all.
Quadratic sequences
When the second differences are constant, the rule contains an n² term, and the coefficient of n² is half the second difference.
For 2, 5, 10, 17, the second difference is 2, so the rule contains 1n². Subtracting n² from each term gives 1, 1, 1, 1 — a constant — so the rule is n² + 1.
That subtraction step is the general method: take the n² part away and whatever remains is an ordinary arithmetic sequence, which you already know how to handle.
The special sequences worth recognising
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 — each term adds one more than the last.
Fibonacci-type: each term is the sum of the two before it, as in 1, 1, 2, 3, 5, 8. These have no simple nth term rule at GCSE, and questions ask you to continue them or to work backwards to a missing earlier term.
Worked examples
Example 1: Finding an arithmetic nth term
Find the nth term of 5, 8, 11, 14, …
The differences are 3, 3, 3, so the sequence is arithmetic and the rule begins 3n.
At n = 1, 3n gives 3, but the first term is 5. The difference is +2, so the rule is 3n + 2.
Check at n = 4: 3(4) + 2 = 14. ✓
Example 2: Testing membership
Is 100 a term of the sequence with nth term 3n + 2?
Set 3n + 2 = 100. Then 3n = 98, so n = 32.67 to two decimal places.
A position must be a positive whole number, so 100 is not a term of this sequence.
Note what the answer to this question actually is: a yes or no with a reason, not a value of n. Writing "n = 32.67" alone does not answer what was asked.
Example 3: A geometric sequence
A geometric sequence begins 4, 12, 36. Find the common ratio and the 6th term.
Divide a term by the one before it: 12 ÷ 4 = 3, and 36 ÷ 12 = 3. The common ratio is 3.
The 6th term is the first term multiplied by the ratio five times: 4 × 3⁵ = 4 × 243 = 972.
Using 3⁶ would give 2916, which is the commonest error here. The first term has been multiplied zero times, so the power is always one less than the position.
Common mistakes and how to avoid them
Using the first term as the coefficient of n. The coefficient is the common difference. For 5, 8, 11 the rule starts 3n, not 5n.
Getting the constant wrong. Substitute n = 1 into your rule and compare with the actual first term before writing it down.
Sign errors in decreasing sequences. A falling sequence has a negative coefficient of n, so 20, 17, 14 gives 23 − 3n.
Confusing difference with ratio. Arithmetic sequences add; geometric sequences multiply. Check both before deciding.
Using the wrong power in a geometric sequence. The nth term is the first term multiplied by the ratio (n − 1) times.
Answering "is this a term?" with a decimal. The decimal is the working; the answer is that the number is not in the sequence.
Forgetting to halve the second difference. In a quadratic sequence, a second difference of 6 gives 3n², not 6n².
Exam technique for "Sequences and the nth Term"
Write the differences underneath the sequence as your first line. It identifies the type, hands you the coefficient of n, and is often worth a method mark on its own.
State the common difference or ratio explicitly before building the rule, so the examiner can see where the coefficient came from.
Check your nth term on a term other than the first — usually the last one given. A rule that fits the first term by construction may still be wrong.
When asked whether a number is in a sequence, finish with a sentence. The value of n is the working, not the answer.
Read the position carefully in geometric questions: the 6th term uses the ratio to the power 5.
If a sequence does not have a constant difference, take second differences before assuming it is geometric. Quadratic sequences are common on the higher tier.
Quick revision summary
Write the differences first. Constant difference means arithmetic; constant ratio means geometric; constant second difference means quadratic.
For an arithmetic sequence, the coefficient of n is the common difference. Then substitute n = 1 and adjust the constant so the rule gives the actual first term. So 5, 8, 11, 14 gives 3n + 2, and 20, 17, 14 gives 23 − 3n.
Use the rule to find any term by substituting its position, and to test membership by solving for n — the number is a term only if n is a positive whole number.
In a geometric sequence, the ratio is found by dividing a term by the one before. The nth term is the first term multiplied by the ratio (n − 1) times. A ratio between 0 and 1 gives a decreasing sequence; a negative ratio gives alternating signs.
In a quadratic sequence, the coefficient of n² is half the second difference. Subtract that n² part from every term and what remains is an ordinary arithmetic sequence.
Recognise the standard ones: squares (n²), cubes (n³), triangular numbers, and Fibonacci-type sequences where each term is the sum of the two before it.