Nth Terms of Linear Sequences

## Nth Terms of Linear Sequences

The value of a given term of a linear sequence can be worked out using the general term of the sequence. The general term is also known as the nth term, because, for any term n in the sequence, the value of the term can be worked out.

To work out what the nth term of a sequence is, work out the common difference.

Next, find out the zeroth term by subtracting the difference from the value of the first term.

Any term can now be worked out using

difference x term + zeroth term value.

This is normally written with term as an n, for example 3n + 1.

For any term, its value can be worked out by substituting the n with the term number.

For the value of the 30th term in the sequence 3n + 1: replace n with 30 and calculate 3 x 30 + 1 = 91.

## Example 1

What is the general term (nth term) for the sequence 2, 0, -2, -4?

Set out a table:

 Term 0 1 2 3 4 ... Value 4 2 0 -2 -4 ... Difference -2 -2 -2 -2 ...

The difference is -2, and the value of the 0th term is 4. The nth term is -2n + 4

## Example 2

Is 32 a number in the sequence given by 3n - 5?

Set the question as an equation: 32 = 3n - 5.

37 = 3n (added 5 to both sides)

12.33 = n (divided both sides by 3)

Because 12.33 is not a whole number it cannot be a term; therefore 32 is not in the sequence.