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. Given the sequence 4, 7, 10, 13, ..., work out the difference between each value:

Term1 234...
Value4 71013...
Difference3 33...

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

Term01 234...
Value14 71013...
Difference33 33...

Any term can now be worked out:

difference x term + zeroth term value.

The nth term is 3n + 1 (3 is the difference, and 1 is the value of the zeroth term).

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.


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

Answer: -2n + 4

Set out a table:

Term01 234...
Value42 0-2-4...
Difference-2-2 -2-2...

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

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

Answer: No

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.