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 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?