# Arithmetic Sequences and Series Sums

## Sequence

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

1, 3, 5, 7, …

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 called a constant.

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

Example

2, 4, 6, 8, 10, 12, 14, 16, 18, …

This sequence has a difference of 2 between each number.

In General we could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, … }

where:

• a is the first term
• d is the difference between the terms (called the “common difference”)
Examples

2, 5, 8, 11, 14, 17, 20, 23, …

Has:

a = 2 (the first term)

d = 3 (the “common difference” between terms)

And we get:

{a, a+d, a+2d, a+3d, … }

So

{2, 2+3, 2+2×3, 2+3×3, … }

{2, 5, 8, 11, … }

## Rule

We can write an Arithmetic Sequence as a rule:

xn = a + d(n−1)

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

Example

Write a rule, and calculate the 10th term, for this Arithmetic Sequence:

5, 10, 15, 20, 25, …

This sequence has a difference of 5 between each number.

The values of a and d are:

a = 5 (the first term)

d = 5 (the “common difference”)

Using the Arithmetic Sequence rule:

xn = a + d(n−1)

xn = 5 + 5(n−1)

xn = 5 + 5n − 5

xn = 5n

So the 10th term is:

x10 = 5×10

x10 = 50

## Summing an Arithmetic Series

To sum up the terms of this arithmetic sequence:

a + (a+d) + (a+2d) + (a+3d) + …

use this formula:

Here is how to use it:

Examples

Add up the first 10 terms of the arithmetic sequence:

{ 1, 5, 9, 13, 17, … }

The values of a, d and n are:

a = 1 (the first term)

d = 4 (the “common difference” between terms)

n = 10 (how many terms to add up)

So:

By subtitute the value of a, d, & n, we get