- 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 *n*th 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 *n*th term of a geometric sequence is: