Sequences

Find and use the nth term of arithmetic, geometric and simple quadratic sequences.

Also known as: nth term, arithmetic sequences, geometric sequences, quadratic sequences.

GCSE Maths · Algebra IGCSE Maths · Algebra and graphs

Key ideas

Arithmetic: common difference $d$ ; $n$ th term $= dn + (a - d)$ .
Geometric: common ratio $r$ ; $n$ th term $= ar^{n-1}$ .
Quadratic sequences have a constant second difference $2a$ in $an^2 + bn + c$ .
To test if a value is a term, set the $n$ th term equal to it; $n$ must be a positive integer.

Arithmetic sequences

$5, 8, 11, 14, \ldots$ goes up by $3$ , so the $n$ th term starts with $3n$ . When $n = 1$ , $3n = 3$ , but the first term is $5$ , so add $2$ : the $n$ th term is $3n + 2$ .

Is it in the sequence?

Is $100$ a term of $6n - 2$ ? Solve $6n - 2 = 100$ : $n = 17$ . Because $17$ is a positive whole number, $100$ is the $17$ th term. If $n$ were not an integer, the value would not be in the sequence.

Geometric and quadratic sequences

A geometric sequence multiplies by a constant ratio: $3, 6, 12, \ldots$ has $n$ th term $3 \times 2^{n-1}$ .

A quadratic sequence has a constant second difference. For $2, 6, 12, 20, 30$ the first differences are $4, 6, 8, 10$ and the second difference is $2$ , so the sequence starts with $1n^2$ ; comparing gives $n^2 + n$ .

Worked examples

Example 1 · nth term

Find the $n$ th term of $5, 8, 11, 14, \ldots$

Step 1.Common difference = 3, so start with 3n.
Step 2.3n gives 3, 6, 9, …; each term is 2 more.

Answer: $3n + 2$

Example 2 · Geometric term

A geometric sequence has first term 3 and common ratio 2. Find the 8th term.

Step 1. $a r^{n-1} = 3 \times 2^{7}$ .
Step 2. $3 \times 128 = 384$ .

Answer: 384

Common mistakes

Writing $n + 3$ for a sequence that goes up in $3$ s (the coefficient of $n$ is the difference).
Using $ar^n$ instead of $ar^{n-1}$ for geometric sequences.
Accepting a non-integer n as evidence that a number is in a sequence.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1SequencesEasier

What is the next term in the sequence $3, 7, 11, 15, \ldots$ ?

Question 2SequencesEasier

Which expression is the $n$ th term of $5, 8, 11, 14, \ldots$ ?