A9AlgebraFoundation & Higher

Sequences

A sequence is a list of numbers that follow a rule. The nth term is a formula that lets you jump straight to any term — the 100th term without writing out the first 99.

40 min Video by Maths Genie AQA GCSE Maths

Sequences and Finding the Nth TermWatch the walkthrough, then read the notes below.

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What you'll learn

Continue and describe sequences
Find the nth term of a linear sequence
Use the nth term to find any term
Recognise special sequences

The nth term of a linear sequence

A linear sequence goes up by the same amount each time (the common difference). The nth term is $(common difference) \times n + (zeroth term)$ .

nth term = d n + (a - d)

d = common difference, a = first term.

1Common difference is 4, so start with

4 n

2First term is 3, but

4 \times 1 = 4

, so subtract 1:

4 n - 1

Answer

4 n - 1

Tip — To check, substitute n = 1, 2, 3 and make sure you get the original sequence back.

Special sequences

Watch for $square numbers$ ( $1, 4, 9, 16, \dots$ ), $cube numbers$ ( $1, 8, 27, \dots$ ), $triangular numbers$ ( $1, 3, 6, 10, \dots$ ) and the $Fibonacci$ sequence (add the two previous terms).

Remember these

nth term = d n + (a - d)

Linear sequence (d = common difference).

1, 4, 9, 16 \to n^{2}

Square numbers.

Watch out for these

Using the first term as the constant: nth term of 3,7,11 = 4n + 3.

It is 4n − 1 (subtract the common difference from the first term).

Thinking the nth term gives the position, not the value.

Substitute the position n to get that term’s value.

Key takeaways

Linear nth term: (common difference)n + adjustment.
Substitute n to find any term; check with n = 1, 2, 3.
Learn the square, cube, triangular and Fibonacci patterns.

Test yourself

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