Sequences and Nth Term

Sequences are patterns of numbers that follow a rule, and finding the nth term is one of the most satisfying skills in GCSE Maths. This topic appears on both Foundation and Higher tier papers and is tested by AQA, Edexcel, and OCR every year. You might be asked to continue a sequence, find the nth term rule for a linear (arithmetic) sequence, or determine whether a given number belongs to a sequence. On this page you will learn how to identify arithmetic sequences, find their nth term formula, and use it to answer typical exam questions. Once you have the method down, it becomes almost mechanical.

What Is a Sequence?

A sequence is an ordered list of numbers that follows a pattern. Each number in the list is called a term. The first term is often written as a₁, the second as a₂, and so on.

An arithmetic sequence (also called a linear sequence) is one where the difference between consecutive terms is constant. This constant gap is called the common difference, usually written as d.

Examples:

3, 7, 11, 15, 19, ... (common difference = 4)
20, 17, 14, 11, 8, ... (common difference = −3)

nth term = dn + (a − d), where d is the common difference and a is the first term

This can also be written as:

nth term = a + (n − 1)d

Both forms are equivalent. The first is often quicker to use at GCSE.

Step-by-Step Method

Finding the nth term of a linear sequence

Work out the common difference (d) by subtracting consecutive terms. For 5, 8, 11, 14: d = 8 − 5 = 3.
Write the "times table" part. The nth term starts with dn, so here it starts with 3n.
Compare 3n with the sequence. When n = 1, 3n = 3, but the first term is 5. The difference is 5 − 3 = 2.
Add the adjustment. The nth term = 3n + 2.
Check with another term. When n = 3, 3(3) + 2 = 11 ✓.

Using the nth term

To find the 50th term: substitute n = 50 into the formula.
To check if a number is in the sequence: set the formula equal to that number, solve for n, and check whether n is a positive whole number.

Generating a sequence from a rule

If given a formula like T(n) = 4n − 7, substitute n = 1, 2, 3, ... to generate the terms: −3, 1, 5, 9, ...

Worked Example 1 — Foundation Level

Question: Find the nth term of the sequence 2, 9, 16, 23, 30, ...

Working:

Step 1: Common difference d = 9 − 2 = 7.

Step 2: The nth term starts with 7n.

Step 3: When n = 1, 7(1) = 7, but the first term is 2. Adjustment = 2 − 7 = −5.

Step 4: nth term = 7n − 5.

Check: n = 4 → 7(4) − 5 = 23 ✓

Answer: nth term = 7n − 5

Worked Example 2 — Higher Level

Question: The nth term of a sequence is 4n + 3. Is 95 a term in this sequence? If so, which term?

Working:

Step 1: Set 4n + 3 = 95.

Step 2: Subtract 3: 4n = 92.

Step 3: Divide by 4: n = 23.

Since n = 23 is a positive whole number, 95 is in the sequence.

Answer: Yes, 95 is the 23rd term.

Follow-up: Is 100 in this sequence?

4n + 3 = 100 → 4n = 97 → n = 24.25

Since n is not a whole number, 100 is not in the sequence.

Common Mistakes

Getting the sign of the adjustment wrong. If d = 5 and the first term is 3, then 5n gives 5 when n = 1, so the adjustment is 3 − 5 = −2, not +2. Always subtract the value of dn at n = 1 from the actual first term.
Confusing the common difference with the first term. The common difference is found by subtraction between consecutive terms, not by looking at the first number.
Forgetting that n must be a positive integer. When checking membership, a decimal or negative value of n means the number is not in the sequence.
Assuming all sequences are arithmetic. If the differences between terms are not constant, the sequence is not linear — it might be quadratic or geometric. Check before applying the linear formula.
Miscounting term positions. The first term corresponds to n = 1, not n = 0.

Exam Tips

Always show how you found d. Write the subtraction explicitly, e.g., "d = 9 − 5 = 4." This earns the first method mark.
Check your formula against at least two terms from the original sequence. If both match, your formula is correct.
"Is x in the sequence?" questions are very common. Set the formula equal to x, solve for n, and state clearly whether n is a whole number.
On Higher papers, you may meet decreasing sequences where d is negative. The method is identical; just be careful with signs.

Practice Questions

Q1 (Foundation): Find the nth term of the sequence 6, 10, 14, 18, 22, ...

Answer: 4n + 2

Q2 (Foundation/Higher): The nth term of a sequence is 3n + 7. Find the 40th term.

Answer: 127

Q3 (Higher): The nth term of a sequence is 6n − 11. Is 200 a term in this sequence? Show working.

Answer: 6n − 11 = 200 → 6n = 211 → n = 35.1̅6̅. Since n is not a whole number, 200 is not in the sequence.

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Summary

A sequence is an ordered list of numbers following a rule.
An arithmetic (linear) sequence has a constant common difference, d.
The nth term of a linear sequence is dn + (first term − d).
Use the formula to find any term, or to check whether a value belongs to the sequence.
Always verify your nth term rule by substituting at least two values of n.
Decreasing sequences simply have a negative common difference.
This topic connects directly to quadratic sequences, which extend the same ideas to second differences.

Sequences and Nth Term –