Difference between revisions of "Arithmetic sequence"

(Definition)
(Sums of Arithmetic Sequences)
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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. For example,
+
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

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. Formally, $s_n=\frac{n}{2}(a_1+a_n)$. For example,

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

or

$\frac{7}{2}(5+17)=77$

Example Problems and Solutions

Introductory Problems


See Also

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