Difference between revisions of "Arithmetic sequence"
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==Definition== | ==Definition== | ||
| − | An arithmetic sequence is a sequence of numbers | + | 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 <math>a_n</math> is defined [[recursive|recursively]] by a first term <math>a_0</math> and <math>a_n = a_{n-1} + d</math> for <math>n \geq 1</math>, where <math>d</math> is the common difference. |
==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. For example, | |
| − | <math>\displaystyle 5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} | + | <math>\displaystyle 5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77</math> |
==See Also== | ==See Also== | ||
| − | [[geometric sequence|Geometric Sequences]] | + | *[[sequence|Sequence]] |
| + | *[[series|Series]] | ||
| + | *[[geometric sequence|Geometric Sequences]] | ||
Revision as of 10:14, 23 June 2006
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.
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,
