Let's learn how to use arithmetic sequence with some examples in A-Level Maths!

**Sequence **

A sequence is a set of things (usually numbers) that are in order.

Each number in the sequence is called a term (or sometimes "element" or "member")

**Arithmetic Sequence **

In an Arithmetic Sequence the difference between one term and the next is a constant.

In other words, we just add the same value each time ... infinitely.

In General we could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }

where:

- a is the first term, and
- d is the difference between the terms (called the "common difference")

Example:

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

- a = 1 (the first term)
- d = 3 (the "common difference" between terms)

And we get:

{a, a+d, a+2d, a+3d, ... }

{1, 1+3, 1+2×3, 1+3×3, ... }

{1, 4, 7, 10, ... }

**Rule **

We can write an Arithmetic Sequence as a rule:

**x**_{n}** = a + d(n−1)**

(We use "n−1" because d is not used in the 1st term).

### Example: Write a rule, and calculate the 9th term, for this Arithmetic Sequence:

3, 8, 13, 18, 23, 28, 33, 38, ...

This sequence has a difference of 5 between each number.

The values of a and d are:

- a = 3 (the first term)
- d = 5 (the "common difference")

Using the Arithmetic Sequence rule:

x_{n} = a + d(n−1)

= 3 + 5(n−1)

= 3 + 5n − 5

= 5n − 2

So the 9th term is:

x_{9} = 5×9 − 2

= 43

**Total** **of first 𝒏** **terms of series**

### Example: Add up the first 10 terms of the arithmetic sequence:

{ 1, 4, 7, 10, 13, ... }

The values of a, d and n are:

- a = 1 (the first term)
- d = 3 (the "common difference" between terms)
- n = 10 (how many terms to add up)

= 5(2+9·3)

= 145

...and if last term is unknown

Drafted by Eunice (Maths)

Reference

https://www.mathsisfun.com/algebra/sequences-sums-arithmetic.html