If the nth term of a sequence is known, it is possible to work out any number in that sequence.

Write the first five terms of the sequence .

represents the position in the sequence. The first term in the sequence is when , the second term in the sequence is when , and so on.

To find the terms, substitute for the position number:

- when ,
- when ,
- when ,
- when ,
- when ,

The first five terms of the sequence: are

The nth term for a quadratic sequence has a term that contains . Terms of a quadratic sequence can be worked out in the same way.

Write the first five terms of the sequence .

- when ,
- when ,
- when ,
- when ,
- when ,

The first five terms of the sequence: are -1, 5, 13, 23, 35

When the nth term is known, it can be used to work out specific terms in a sequence. For example, the 50th term can be calculated without calculating the first 49 terms, which would take a long time.

- Question
What is the 100th term in the sequence ?

To answer this, the position is 100, so substitute for 100.

497 is the 100th term in the sequence .

- Question
Is the number 14 in the sequence ?

To work out whether 14 is in this sequence, put the nth term equal to the number and solve the equation.

This means that 14 is in the sequence and it is the third term.