# Difference between revisions of "Arithmetico-geometric series"

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== Infinite Sum == | == 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>. | + | 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> (<math>|r|<1</math>). 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>S=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots</math> |

## 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 (). Or, , where is the infinite sum of the .