Difference between revisions of "Arithmetico-geometric series"
m (→Finite Sum: fixed error: g_1 missing after a_1 in 2 places) |
m (→Infinite Sum: fixed error: added missing g_1 after a_1) |
||
Line 25: | Line 25: | ||
<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> | ||
− | <math>rS=a_1g_1r+( | + | <math>rS=a_1g_1r+(a_1g_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>rS-S=-a_1g_1-dg_1r-dg_1r^2-dg_1r^3-\ldots=-a_1g_1+\frac{dg_1r}{r-1}</math> |
Revision as of 02:27, 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.
Contents
[hide]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 .