Difference between revisions of "Arithmetic sequence"
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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. Explicitly, it can be defined as <math>a_n=a_0+dn</math>. | 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. Explicitly, it can be defined as <math>a_n=a_0+dn</math>. | ||
| + | ==Terms in an Arithmetic Sequence== | ||
| + | To find the <math>n^{th} </math> term in an arithmetic sequence, you use the formula | ||
| + | <cmath>a_n = a_1 + d(n-1)</cmath> | ||
| + | where <math>a_n</math> is the <math>n^{th}</math> term, <math>a_1</math> is the first term, and <math>d</math> is the difference between consecutive terms. | ||
==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. Formally, <math>s_n=\frac{n}{2}(a_1+a_n)</math>. 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> | + | <math>5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77</math> |
or | or | ||
Revision as of 14:31, 15 August 2015
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 
.
Terms in an Arithmetic Sequence
To find the 
term in an arithmetic sequence, you use the formula
![\[a_n = a_1 + d(n-1)\]](http://latex.artofproblemsolving.com/7/f/2/7f2b52ffef55f17744e75c96ec9c9ce85fa69f31.png)
whereis the
term,
is the first term, and
is the difference between consecutive terms.
is the first term, and 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
Intermediate Problems
- Find the roots of the polynomial
, given that the roots form an arithmetic progression.
, given that the roots form an arithmetic progression.
