Difference between revisions of "Arithmetic sequence"
ComplexZeta (talk | contribs) m (→Definition) |
(started example problem section) |
||
| Line 1: | Line 1: | ||
==Definition== | ==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 <math>a_n</math> is defined [[recursion|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. | 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 [[recursion|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== | ||
| Line 7: | Line 8: | ||
<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> | ||
| + | |||
| + | |||
| + | == Example Problems and Solutions == | ||
| + | === Introductory Problems === | ||
| + | * [[2005_AMC_10A_Problems/Problem_17 | 2005 AMC 10A Problem 17]] | ||
| + | |||
==See Also== | ==See Also== | ||
Revision as of 11:34, 2 August 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.
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,
Example Problems and Solutions
Introductory Problems
