# Difference between revisions of "Arithmetic sequence"

(→Definition) |
(→Sums of Arithmetic Sequences) |
||

Line 4: | Line 4: | ||

==Sums of Arithmetic Sequences== | ==Sums of Arithmetic Sequences== | ||

− | There are many ways of calculating the sum of the terms of a [[finite]] arithmetic sequence. Perhaps the simplest is to take the average, or [[arithmetic mean]], of the first and last term and to multiply this by the number of terms. | + | There are many ways of calculating the sum of the terms of a [[finite]] arithmetic sequence. Perhaps the simplest is to take the average, or [[arithmetic mean]], of the first and last term and to multiply this by the number of terms. Formally, <math>s_n=\frac{n}{2}(a_1+a_n)</math>. For example, |

− | <math>\displaystyle 5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77</math> | + | <math>\displaystyle 5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77</math> |

+ | or | ||

+ | |||

+ | <math>\frac{7}{2}(5+17)=77</math> | ||

== Example Problems and Solutions == | == Example Problems and Solutions == |

## Revision as of 21:56, 4 November 2006

## Contents

## Definition

An **arithmetic sequence** is a sequence of numbers in which each term is given by adding a fixed value to the previous term. For example, -2, 1, 4, 7, 10, ... is an arithmetic sequence because each term is three more than the previous term. In this case, 3 is called the *common difference* of the sequence. More formally, an arithmetic sequence is defined recursively by a first term and for , where is the common difference. Explicitly, it can be defined as .

## Sums of Arithmetic Sequences

There are many ways of calculating the sum of the terms of a finite arithmetic sequence. Perhaps the simplest is to take the average, or arithmetic mean, of the first and last term and to multiply this by the number of terms. Formally, . For example,

or

## Example Problems and Solutions

### Introductory Problems