# Difference between revisions of "Arithmetico-geometric series"

m (→Infinite Sum: Changed from difference to ratio.) |
m (→Finite Sum: fixed error: g_1 missing after a_1 in 2 places) |
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<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+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+( | + | <math>S_n=a_1g_1+(a_1g_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+( | + | <math>rS_n=a_1g_1r+(a_1g_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>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> |

## Revision as of 03:26, 24 July 2012

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