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>.
  
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==Terms in an Arithmetic Sequence==
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To find the <math>n^{th} </math> term in an arithmetic sequence, you use the formula
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<cmath>a_n = a_1 + d(n-1)</cmath>
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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>\displaystyle 5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77</math>
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<math>5 + 7 + 9 + 11 + 13 + 15 + 17 = \frac{5+17}{2} \cdot 7 = 77</math>
  
 
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Revision as of 14:31, 15 August 2015

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 $a_n=a_0+dn$.

Terms in an Arithmetic Sequence

To find the $n^{th}$ term in an arithmetic sequence, you use the formula \[a_n = a_1 + d(n-1)\]

where $a_n$ is the $n^{th}$ term, $a_1$ is the first term, and $d$ is the difference between consecutive terms.

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,

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

or

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

Example Problems and Solutions

Introductory Problems

Intermediate Problems

  • Find the roots of the polynomial $x^5-5x^4-35x^3+ax^2+bx+c$, given that the roots form an arithmetic progression.

See Also

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