Difference between revisions of "Arithmetic sequence"
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==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 | ||
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| + | <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
