# Difference between revisions of "Arithmetic sequence"

## 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 $a_n$ is defined recursively by a first term $a_0$ and $a_n = a_{n-1} + d$ for $n \geq 1$, where $d$ is the common difference. Explicitly, it can be defined as $a_n=a_0+dn$.

## 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. For example,

$\displaystyle 5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77$

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