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• A-Level Maths

# Sequences & Series

A sequence is a list of numbers with a particular relation; a series is the sum of such a list of numbers. Sequences are sometimes a,so referred to as progressions.

## Arithmetic

### Arithmetic Sequences

An arithmetic sequence has a constant defined distance between terms, e.g. 1, 3, 5 7, 9 etc. The first term is 1 and the common difference is +2.

The common difference can be positive (the sequence is increasing) or negative (the series is decreasing).

To calculate the nth term, u(n), of an arithmetic sequence:

where a is the first term and d the common difference.

### Arithmetic Series

An arithmetic series is the sum of all numbers in an arithmetic sequence. The sum of the first n terms is given by:

where a is the first term, d the common difference, and l the last term.

## Geometric

### Geometric Sequences

A geometric sequence is a sequence where there is a common ratio, not a common difference. This means the relationship between the numbers is a multiplication, not addition/subtraction. For example, the sequence 2, 4, 8, 16, 32 etc. is geometric - each term is multiplied by 2.

The formula for the nth term of a geometric sequence is: