GCSE — mathsFoundation2 min read

Sequences and the Nth Term

Find the nth term of arithmetic sequences for GCSE Maths. Generate terms and identify sequences from patterns.

Tutor AI Team2026-02-25T01:45:43.857Z

Sequences are ordered lists of numbers following a pattern. The nth term formula lets you find any term without listing all previous terms.

Core Concepts

Arithmetic Sequences

An arithmetic sequence has a common difference $d$ between consecutive terms.

$3, 7, 11, 15, ...$ → common difference $d = 4$

The Nth Term Formula

$T_n = a + (n-1)d$

or equivalently $T_n = dn + (a - d)$

where $a$ = first term, $d$ = common difference.

Finding the Nth Term

Step 1: Find the common difference $d$ . Step 2: The nth term is $dn + c$ where $c = a - d$ .

Example: $5, 8, 11, 14, ...$

$d = 3$ . Nth term = $3n + c$ . When $n=1$ : $3(1) + c = 5$ → $c = 2$ .

Nth term = $3n + 2$ .

Using the Nth Term

Find the 50th term: substitute $n = 50$ .
Is 100 in the sequence? Set $3n + 2 = 100$ and check if $n$ is a positive integer.

Worked Example: Example 1

Problem

$2, 9, 16, 23, ...$ → $d = 7$ . Nth term = $7n - 5$ .

Solution

Worked Example: Example 2

Problem

Is 45 in the sequence $4, 7, 10, 13, ...$ ?

$3n + 1 = 45$ → $n = 14.\overline{6}$ . Not an integer → 45 is NOT in the sequence.

Solution

Practice Problems

1. Find the nth term: $6, 10, 14, 18, ...$
1. Find the 20th term of $3, 8, 13, 18, ...$
1. Is 200 in the sequence $7, 12, 17, 22, ...$ ?

Want to check your answers and get step-by-step solutions?

Key Takeaways

✓
Common difference $d$ = difference between consecutive terms.
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Nth term = $dn + (a - d)$ .
✓
To check membership, solve for $n$ and check if it's a positive integer.

Tags:

#gcse#maths#algebra#sequences#nth-term

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