# Difference between revisions of "Arithmetico-geometric series"

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Let <math>S_n</math> represent the sum of the first n terms. | Let <math>S_n</math> represent the sum of the first n terms. | ||

− | <math>S_n=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r)+\ldots+(a_1+(n-1)d)(g_1r^{n-1})</math> | + | <math>S_n=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots+(a_1+(n-1)d)(g_1r^{n-1})</math> |

<math>S_n=a_1g_1+(a_1+dg_1)r+(a_1g_1+2dg_1)r^2+\ldots+(a_1g_1+(n-1)dg_1)r^{n-1}</math> | <math>S_n=a_1g_1+(a_1+dg_1)r+(a_1g_1+2dg_1)r^2+\ldots+(a_1g_1+(n-1)dg_1)r^{n-1}</math> | ||

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<math>S_n=\frac{(a_1+(n-1)d)g_1r^n}{r-1}-\frac{a_1g_1}{r-1}-\frac{dg_1r(r^{n-1}-1)}{(r-1)^2}=\frac{a_ng_{n+1}}{r-1}-\frac{x_1}{r-1}-\frac{d(g_{n+1}-g_2)}{(r-1)^2}</math> | <math>S_n=\frac{(a_1+(n-1)d)g_1r^n}{r-1}-\frac{a_1g_1}{r-1}-\frac{dg_1r(r^{n-1}-1)}{(r-1)^2}=\frac{a_ng_{n+1}}{r-1}-\frac{x_1}{r-1}-\frac{d(g_{n+1}-g_2)}{(r-1)^2}</math> | ||

+ | |||

+ | == Infinite Sum == | ||

+ | The sum of an infinite arithmetico-geometric sequence is <math>\frac{dg_2}{(1-r)^2}+\frac{x_1}{1-r}</math>, where <math>d</math> is the common difference of <math>a_n</math> and <math>r</math> is the common difference of <math>g_n</math> (|r|<1). Or, <math>\frac{drS_g+x_1}{1-r}</math>, where <math>S_g</math> is the infinite sum of the <math>g_n</math>. | ||

+ | |||

+ | <math>S=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots</math> | ||

+ | |||

+ | <math>rS=a_1g_1r+(a_1+dg_1)r^2+(a_1g_1+2dg_1)r^3+.\,.\,.</math> | ||

+ | |||

+ | <math>rS-S=-a_1g_1-dg_1r-dg_1r^2-dg_1r^3-\ldots=-a_1g_1+\frac{dg_1r}{r-1}</math> | ||

+ | |||

+ | <math>S=\frac{dg_1r}{(r-1)^2}-\frac{a_1g_1}{r-1}=\frac{dg_2}{(r-1)^2}-\frac{x_1}{r-1}=\frac{dg_2}{(1-r)^2}+\frac{x_1}{1-r}</math> | ||

== Example Problems == | == Example Problems == | ||

* [[Mock_AIME_2_2006-2007/Problem_5 | Mock AIME 2 2006-2007 Problem 5]] | * [[Mock_AIME_2_2006-2007/Problem_5 | Mock AIME 2 2006-2007 Problem 5]] |

## Revision as of 23:37, 4 November 2006

An arithmetico-geometric series is the sum of consecutive terms in an arithmetico-geometric sequence defined as: , where and are the th terms of arithmetic and geometric sequences, respectively.

## Finite Sum

The sum of the first n terms of an arithmetico-geometric sequence is , where is the common difference of and is the common ratio of . Or, , where is the sum of the first terms of .

**Proof:**

Let represent the sum of the first n terms.

## Infinite Sum

The sum of an infinite arithmetico-geometric sequence is , where is the common difference of and is the common difference of (|r|<1). Or, , where is the infinite sum of the .