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2.3 Using the nth term

2.3 Using the nth term

You already know how to work out the position-to-term rule of a linear sequence.
Example: The sequence 5, 7, 9, 11, …, …
has position-to-term rule: term = 2 × position number + 3

You can also write the position-to-term rule called the nth term.
To do this, you replace the words ‘position number’ with the letter n. n stands for the term number.

So, in the example above,
instead of writing term = 2 × position number + 3
you write: nth term is 2 × n + 3 2 × n is usually written as 2n.
or simply: nth term is 2n + 3

Worked example 2.3

The nth term of a sequence is 2n − 1. Work out the first three terms and the ninth term of the sequence.

1st term = 2 × 1 − 1 = 1 To find the first term, substitute n = 1 into the expression.
2nd term = 2 × 2 − 1 = 3 To find the second term, substitute n = 2 into the expression.
3rd term = 2 × 3 − 1 = 5 To find the third term, substitute n = 3 into the expression.
9th term = 2 × 9 − 1 = 17 To find the ninth term, substitute n = 9 into the expression.

✦ Exercise 2.3

1 Work out the first three terms and the 10th term of the sequences with the given nth term.
a n + 6 b n − 3 c 4n d 6n
e 2n + 5 f 3n − 1 g 5n + 3 h 4n − 3
2 This pattern is made from pink squares.
Pattern 1 Pattern 2 Pattern 3 Pattern 4

a Write down the sequence of the numbers of pink squares.
b Write down the term-to-term rule.
c Explain how the sequence is formed.
d Work out the position-to-term rule.
e Copy and complete the working to show that the nth term is 3n + 1.
first term = 3 × 1 + 1 = 4 second term = 3 × + 1 =
third term = 3 × + 1 = fourth term = 3 × + 1 =

3 This pattern is made from dots.
Pattern 1 Pattern 2 Pattern 3 Pattern 4

Hassan thinks that the nth term for the sequence of numbers of dots is 2n + 3.
Is Hassan correct? Explain how you worked out your answer.

2 Sequences, expressions and formulae 23