1. Arithmetic Progressions

by M. Bourne


We want a sequence of numbers. Let's start with a number: a1.

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

We get a1 + d and the first 2 terms in our sequence are:

a1, a1 + d

For the next term, let's add d to that last term and we have a1 + 2d.

Our sequence is now:

a1, a1 + d, a1 + 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

Let's start with a1 = 4 and then add d = 3 each time to get each new number in the sequence. We get:

4, 7, 10, 13, …

The nth term, an of an AP is:

math expression

LiveMath can give us the n-th term of an AP:

LIVEMath


Sum of an Arithmetic Progression

The sum to n terms of an AP is:

math expression


Proofs


LiveMath can give us the sum to n terms of an AP.

LIVEMath


Example 1

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


Answer


Example 2

Find the sum of the first 1000 odd numbers.


Answer


Example 3

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


Answer




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