# Difference between revisions of "Arithmetic series"

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To find the sum of an arithmetic sequence, we can write it out in two as so (<math>S</math> is the sum, <math>a</math> is the first term, <math>z</math> is the number of terms, and <math>d</math> is the common difference): | To find the sum of an arithmetic sequence, we can write it out in two as so (<math>S</math> is the sum, <math>a</math> is the first term, <math>z</math> is the number of terms, and <math>d</math> is the common difference): | ||

<cmath> | <cmath> | ||

− | S = a + (a+d) + (a+2d) + | + | S = a + (a+d) + (a+2d) + \ldots + (z-d) + z |

</cmath> | </cmath> | ||

Flipping the right side of the equation we get | Flipping the right side of the equation we get | ||

<cmath> | <cmath> | ||

− | S = z + (z-d) + (z-2d) + | + | S = z + (z-d) + (z-2d) + \ldots + (a+d) + a |

</cmath> | </cmath> | ||

## Revision as of 20:30, 19 September 2015

An **arithmetic series** is a sum of consecutive terms in an arithmetic sequence. For instance,

is an arithmetic series whose value is 50.

To find the sum of an arithmetic sequence, we can write it out in two as so ( is the sum, is the first term, is the number of terms, and is the common difference): Flipping the right side of the equation we get

Now, adding the above two equations vertically, we get

This equals , so the sum is .

## Contents

## Problems

### Introductory Problems

### Intermediate Problems

### Olympiad Problem

## See also

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