The nth term

Foundation and Higher

The idea

A linear sequence goes up by the same amount $d$ each time, so its $n$ th term is $dn + c$ . For $5, 8, 11, 14$ the difference is $3$ , so it is $3n + 2$ .

Find the common difference $d$ : that is the number in front of $n$ .
Work out what $d \times 1$ needs to become the first term: $3 \times 1 + 2 = 5$ .
Check with the second term: $3 \times 2 + 2 = 8$ .

Watch out The number added is not the first term: $5, 8, 11$ is $3n + 2$ , not $3n + 5$ .

See it worked

Worked exampleCalculator allowed2 marks

The nth term of a sequence is 7n $-$ 4, where the first term has n = 1. Work out the tenth term.

$7\times 10-4$
Substitute n = 10 into the position rule.
7 $\times$ 10 $-$ 4 = 66.

Answer $66$

Your turn

Question 1Non-calculator3 marks

Here are the first four terms of a sequence: $7, 11, 15, 19$ .

(a) Find an expression for the $n$ th term of the sequence.

(b) Work out the 20th term of the sequence.

Question 2Non-calculator4 marks

An arithmetic sequence has third term 15 and eighth term 45.

(a) Find its common difference.

(b) Find the position of the term 99.

Question 3Non-calculator5 marks

Sequence A has $n$ th term $3n + 2$ . Sequence B has $n$ th term $47 - 2n$ . Sequence C has $n$ th term $4n + 1$ .

(a) Sequences A and B have one term in the same position that is equal. Find the value of that term.

(b) Find the three smallest numbers that are terms of both sequence A and sequence C.

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