# Difference between revisions of "Arithmetico-geometric series"

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− | == | + | An arithmetico-geometric series is the sum of consecutive terms in an arithmetico-geometric sequence defined as: <math>x_n=a_ng_n</math>, where <math>a_n</math> and <math>g_n</math> are the <math>n</math>th terms of arithmetic and geometric sequences, respectively. |

+ | == Finite Sum == | ||

+ | The sum of the first n terms of an arithmetico-geometric sequence is <math>\frac{a_ng_{n+1}}{r-1}-\frac{x_1}{r-1}-\frac{d(g_{n+1}-g_2)}{(r-1)^2}</math>, where <math>d</math> is the common difference of <math>a_n</math> and <math>r</math> is the common ratio of <math>g_n</math>. | ||

+ | |||

+ | '''Proof:''' | ||

+ | |||

+ | <math>x_n=(a_1+d(n-1))(g_1\cdot r^{n-1})</math> | ||

+ | |||

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

+ | |||

+ | <math>rS_n=a_1g_1r+(a_1+dg_1)r^2+(a_1g_1+2dg_1)r^3+\ldots+(a_1g_1+(n-1)dg_1)r^{n}</math> | ||

+ | |||

+ | <math>rS_n-S_n=-a_1g_1-dg_1r-dg_1r^2-dg_1r^3-\ldots-dg_1r^{n-1}+(a_1g_1+(n-1)dg_1)r^n</math> | ||

+ | |||

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

+ | |||

+ | == 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 23:00, 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 .

**Proof:**

Let represent the sum of the first n terms.