Arithmetic Sequences Explained: Finding the nth Term for GCSE Maths

Sequences turn up on every GCSE Maths paper, Foundation and Higher. They are usually worth between three and six marks, and the questions follow a tiny number of patterns. If you can find a common difference and write an nth term rule, you can bank those marks in under two minutes. This guide walks through the whole topic, from spotting the pattern to summing a series, using the same worked examples a tutor would put in front of you.

Arithmetic vs geometric: spot the pattern first

Before you do anything with a sequence, decide what type it is. There are two you need to recognise at GCSE.

Sequence	Pattern	Type	What stays the same
3, 7, 11, 15, 19, 23, 27	Add 4 each time	Arithmetic (linear)	Common difference, d = 4
3, 6, 12, 24, 48, 96, 192	Multiply by 2 each time	Geometric	Common ratio, r = 2

An arithmetic sequence is built on addition and subtraction. A geometric sequence is built on multiplication and division. That’s the whole distinction.

To find the common difference, subtract any term from the one after it: 7 − 3 = 4, and 11 − 7 = 4. To find a common ratio, divide instead: 6 ÷ 3 = 2, and 12 ÷ 6 = 2.

Tip: Always check two gaps, not one. The sequence 1, 2, 4, 7, 11 has a first gap of 1 and looks arithmetic if you stop there. The second gap is 2, so it isn’t. Exam setters love this trap.

Foundation tier students mostly meet arithmetic sequences. Higher tier students also need to recognise geometric sequences, quadratic sequences and Fibonacci-style sequences, but the arithmetic case is the foundation for all of them. If you want a quick refresher on the surrounding algebra first, our guide to all of GCSE Algebra in one place covers it.

The mean trick: a quick sanity check

Here’s a neat property that doubles as a checking tool. In an arithmetic sequence, the mean of any two terms gives you the term halfway between them.

Take 3 and 11 from the sequence above. Their mean is (3 + 11) ÷ 2 = 7, which is exactly the term sitting between them. Try 7 and 23: (7 + 23) ÷ 2 = 15. Right again.

Geometric sequences do the same thing with the geometric mean, which is the square root of the product. For 3 and 12, √(3 × 12) = √36 = 6, the term in the middle.

You won’t be asked to name the geometric mean at GCSE, but the arithmetic version is a two-second way to check that you’ve copied a sequence down correctly.

The nth term formula

This is the formula that unlocks the topic:

nth term = a + (n − 1)d

where a is the first term, d is the common difference and n is the position of the term you want.

Let’s find the 5th term of 3, 7, 11, 15, …

• a = 3, d = 4, n = 5
• 3 + (5 − 1) × 4 = 3 + 16 = 19

That matches the sequence. The point of the formula is that it works just as well for the 100th term, which you would never want to count up to by hand.

Did you know? The equivalent formula for a geometric sequence is a × r^(n − 1). For 3, 6, 12, 24, … the 6th term is 3 × 2⁵ = 3 × 32 = 96. Higher tier students should have this one in their back pocket too.

Writing the nth term rule in exam form

Exam questions almost never ask for a single term. They ask you to “find an expression for the nth term”, and they want it simplified into the form dn + c. Here is the method with a worked example.

Sequence: 8, 14, 20, 26, …

1. Find d: 14 − 8 = 6 (check: 20 − 14 = 6). So d = 6.
2. Write the formula: 8 + (n − 1) × 6.
3. Expand the bracket: 8 + 6n − 6.
4. Simplify: 6n + 2.
5. Check: n = 1 gives 6 + 2 = 8 ✓. n = 4 gives 24 + 2 = 26 ✓.

Now a decreasing one. Sequence: 150, 143, 136, …

1. d = 143 − 150 = −7.
2. 150 + (n − 1) × (−7) = 150 − 7n + 7.
3. Simplify: 157 − 7n.
4. Check: n = 1 gives 157 − 7 = 150 ✓.

Shortcut: The number in front of n is always the common difference. To find the constant, work out what comes before the first term (the “zeroth term”) by subtracting d once. For 8, 14, 20 the zeroth term is 8 − 6 = 2, so the rule is 6n + 2. Most students find this faster than expanding the bracket, but learn both so you can check one against the other.

Sequences of fractions

Higher tier papers sometimes present a sequence of fractions like this:

2/3, 3/5, 4/7, 5/9, 6/11, …

Don’t panic. Treat the numerators and denominators as two separate arithmetic sequences.

• Numerators: 2, 3, 4, 5, 6 → a = 2, d = 1 → nth term is n + 1
• Denominators: 3, 5, 7, 9, 11 → a = 3, d = 2 → nth term is 2n + 1

Put them back together: the nth term is (n + 1) / (2n + 1). Check with n = 3: (3 + 1)/(6 + 1) = 4/7 ✓.

Recursive (term-to-term) rules

A position-to-term rule like 3n − 7 lets you jump straight to any term. A term-to-term rule tells you how to get the next term from the previous one, and you must work through in order.

Suppose the first term is 3 and each term is “the previous term plus 4”. Then the terms are 3, 7, 11, 15, 19: an arithmetic sequence with d = 4.

Now suppose the first term is 2 and each term is “3 times the previous term, plus 2”. That gives 2, 8, 26, 80, 242. This is neither arithmetic nor geometric, which is a perfectly acceptable answer if a question asks you to classify it.

To write the first few terms from a position-to-term rule, substitute n = 1, 2, 3, 4 in turn. For 3n − 7 that gives −4, −1, 2, 5. Notice d = 3, matching the coefficient of n.

Sequence or series? Finite or infinite?

Two bits of vocabulary that cost easy marks if you mix them up:

• A sequence is a list: 4, 7, 10, 13, 16, 19.
• A series is a sum: 5 + 9 + 13 + 17 + …
• Finite means it stops. Infinite is signalled by the three dots at the end.

So 4, 7, 10, 13, 16, 19 is a finite arithmetic sequence, while 5 + 9 + 13 + … is an infinite arithmetic series.

Summing an arithmetic series: the pairing trick

The formula for the sum of the first n terms is A-level content, so you will not be asked to quote it at GCSE. But the idea behind it is a fantastic problem-solving tool, and it shows up in disguise on non-calculator papers.

Sum of the first n terms = (first term + last term) ÷ 2 × n

In words: find the average of the first and last terms, then multiply by how many terms there are.

Let’s sum 3 + 7 + 11 + 15 + 19 + 23 + 27.

• First term 3, last term 27, seven terms.
• (3 + 27) ÷ 2 = 15, and 15 × 7 = 105.

Add them up on a calculator to confirm. It works because the terms pair up: 3 + 27 = 30, 7 + 23 = 30, 11 + 19 = 30, with 15 in the middle.

A classic challenge question: find the sum of all the even numbers from 2 to 100.

1. First term 2, last term 100, d = 2.
2. How many terms? Use the nth term formula: 2 + (n − 1) × 2 = 100, so n − 1 = 49 and n = 50.
3. Sum = (2 + 100) ÷ 2 × 50 = 51 × 50 = 2,550.

Step 2 is the bit students skip, and it’s where the marks are. Never assume the number of terms; solve for it. For more tricks like this that save time when the calculator is banned, see our post on non-calculator tactics.

Remember: If a problem-solving question involves adding a long run of evenly spaced numbers, you are looking at an arithmetic series. Pair the ends, find the middle, multiply by the count.

Common mistakes to avoid

Mistake	Why it costs marks	Fix
Writing n + 4 instead of 4n	Confuses “add 4 each time” with the rule	The coefficient of n is the common difference
Only checking one gap	Misses quadratic and Fibonacci sequences	Always check at least two gaps
Forgetting a negative d	Turns 157 − 7n into 143 + 7n	Subtract second minus first, keep the sign
Guessing the number of terms	Off-by-one errors in sums	Solve a + (n − 1)d = last term for n
Not checking the rule	A wrong rule loses every follow-on mark	Substitute n = 1 and n = 2 before moving on

Building the habit of checking every rule is exactly the kind of foundation-first approach we describe in the secret study tip to dominate GCSE Maths. It costs ten seconds and saves whole questions.

Practise sequences until the method is automatic

Sequences reward repetition more than almost any other GCSE topic. The method never changes, so the aim is to get it so fast that you can spend your thinking time on the problem-solving questions at the end of the paper.

The GCSE Algebra app has a dedicated sequences section covering common differences, nth term rules, term-to-term rules and geometric sequences, split by Foundation and Higher tier so you only practise what your paper will ask. Every question comes with a worked solution, so when you slip on a negative common difference you see exactly where it went wrong.

If you would rather revise sequences alongside the rest of the syllabus, the GCSE Maths app bundles Number, Algebra, Geometry and Statistics into a single download, giving you 2900+ questions across the four subject apps. Work through the sequences topic, then test yourself with a mixed set to make sure the rule still comes out right when it’s sandwiched between a percentages question and a bearings question.

Ten questions a day for a week and the nth term will be the first mark you bank in the exam hall.