# 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, ...

## General Term of an Arithmetic Progression

The nth 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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