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. | 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 == | == Finite Sum == | ||
− | The sum of the first n terms of an <math>\ | + | The sum of the first n terms of an <math>\textit{arithmetico-geometric sequence}</math> 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>. Or, <math>\frac{a_ng_{n+1}-x_1-drS_g}{r-1}</math>, where <math>S_g</math> is the sum of the first <math>n</math> terms of <math>g_n</math>. |
'''Proof:''' | '''Proof:''' |
Revision as of 19:13, 18 January 2014
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 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 .