# 1. Arithmetic Progressions

by M. Bourne

We want a sequence of numbers. Let's start with a number: `a_1`.

Now add a number `d`, (for "difference").

We get `a_1 + d`* *and the first 2 terms in our sequence are:

`a_1`,

`a_1 + d`

For the next term, let's add another `d` to that last term and we have `a_1 + 2d`.

Our sequence is now:

`a_1`,

`a_1 + d`,

`a_1 + 2d`

We continue this process for as long as we can stay awake. The resulting set of numbers is called an **arithmetic progression** (AP) or **arithmetic sequence**.

### Example 1

Let's start with `a_1 = 4` and then add `d=3` each time to get each new number in the sequence. We get:

`4, 7, 10, 13, ...`

Continues below ⇩

## General Term of an Arithmetic Progression

The *n*th term,
`a_n` of an AP is:

`a_n=a_1+(n-1)d`

## Sum of an Arithmetic Progression

The **sum to** *n***terms** of an AP is:

`S_n=n/2(a_1+a_n)\ or\ ` ` S_n=n/2[2a_1+(n-1)d]`

### Example 2

Using the **second
formula**, find the sum of the first 10 terms for
the series that we met above: `4, 7, 10, 13, ...`

### Example 3

Find the sum of the first
`1000` odd numbers.

### Example 4

A clock strikes the number of times of the hour. How many strikes does it make in one day?

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