# Arithmetic Sequences and Sums

## 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"), read Sequences and Series for more details.

## 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.

### Example: 1, 4, 7, 10, 13, 16, 19, 22, 25, ...

This sequence has a difference of 3 between each number.

The pattern is continued by **adding 3** to the last number each time, like this:

**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 (continued): 1, 4, 7, 10, 13, 16, 19, 22, 25, ...

Has:

- 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**= a + d(n−1)

_{n}**5n − 2**

So the **9**th term is:

x_{9} = 5×**9** − 2

= 43

Is that right? Check for yourself!

Arithmetic Sequences are sometimes called Arithmetic Progressions (A.P.’s)

**Advanced Topic**: Summing an Arithmetic Series

**To sum up** the terms of this arithmetic sequence:

a + (a+d) + (a+2d) + (a+3d) + ...

use this formula:

What is that funny symbol? It is called Sigma Notation

And below and above it are shown the starting and ending values:

It says "Sum up * n* where

*goes from 1 to 4. Answer=*

**n****10**

Here is how to use it:

### 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)

So:

Becomes:

= 5(2+9·3) = 5(29) = 145

*Check: why don't you add up the terms yourself, and see if it comes to 145*

## Footnote: Why Does the Formula Work?

Let's see **why** the formula works, because we get to use an interesting "trick" which is worth knowing.

**First**, we will call the whole sum **"S"**:

**Next**, rewrite S in reverse order:

Now add those two, term by term:

S |
= | a | + | (a+d) | + | ... | + | (a + (n-2)d) | + | (a + (n-1)d) |

S |
= | (a + (n-1)d) | + | (a + (n-2)d) | + | ... | + | (a + d) | + | a |

2S |
= | (2a + (n-1)d) | + | (2a + (n-1)d) | + | ... | + | (2a + (n-1)d) | + | (2a + (n-1)d) |

**Each term is the same!** And there are "n" of them so ...

Now, just divide by 2 and we get:

S = \frac{n}{2} (2a + (n−1)d)

Which is our formula: