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2.2 Finding the nth term

2.2 Finding the nth term

A linear sequence may also be called an arithmetic sequence.
You can write the position-to-term rule of an arithmetic sequence as an expression called the nth term.
For example, the arithmetic sequence 8, 11, 14, 17, ...
has position-to-term rule term = 3 × position number + 5
which can be written as: nth term is 3n + 5.
But when you look at an arithmetic sequence of numbers, how can you work out the expression for the
nth term?
Start by writing the sequence in a table like this. n 1 2 3 4
!e sequence is increasing by 3 every time, so the nth
term expression will start with 3n. terms 8 11 14 17
Add a row for 3n to the table and work out the values. 3n 3 × 1 3 × 2 3 × 3 3 × 4

Now note that you need to add 5 to every 3n term to get 3 6 9 12
the terms in this sequence. 3n + 5 3 + 5 6 + 5 9 + 5 12 + 5
!is means the expression for the nth term is 3n + 5. 8 11 14 17

Worked example 2.2

a The nth term of a sequence is 6 − 2n. Work out the first three terms and the 10th term of the
sequence.
b Work out the expression for the nth term for each sequence.
i 6, 8, 10, 12, ... ii 10, 7, 4, 1, ...

a 1st term = 6 − 2 × 1 = 6 − 2 = 4 To find the first term substitute n = 1 into the expression.
2nd term = 6 − 2 × 2 = 6 − 4 = 2 To find the second term substitute n = 2 into the expression.
3rd term = 6 − 2 × 3 = 6 − 6 = 0 To find the third term substitute n = 3 into the expression.
10th term = 6 − 2 × 10 = 6 − 20 = −14 To find the tenth term substitute n = 10 into the expression.

b i n 1 2 3 4 Start by writing the sequence in a table.
The sequence is increasing by 2 every time, so the
terms 6 8 10 12
nth term expression will start with 2n. Add a row
2n 2 4 6 8 for 2n to the table and work out the values.
2n + 4 6 8 10 12 You can see that you need to add 4 to the 2n values
to get the terms of the sequence.
nth term expression is 2n + 4.
ii n 1 2 3 4 Start by writing the sequence in a table.
The sequence is decreasing by 3 every time, so the
terms 10 7 4 1 nth term expression will start with −3n. Add a row
−3n −3 −6 −9 −12 for −3n to the table and work out the values.
You need to add 13 to the −3n values to get the terms
−3n + 13 10 7 4 1
of the sequence.
nth term expression is −3n + 13.
This can be written as 13 − 3n. It is neater to write the expression with the positive term
before the negative one.

18 2 Sequences and functions