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 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>. 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>. | + | The sum of the first <math>n</math> terms of an <math>\textbf{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:''' | ||
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<math>x_n=(a_1+d(n-1))(g_1\cdot r^{n-1})</math> | <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. | + | Let <math>S_n</math> represent the sum of the first <math>n</math> terms. |
<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> | ||
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<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> | ||
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== Example Problems == | == 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]] | ||
+ | * [[1994_AIME_Problems/Problem_4 | 1994 AIME Problem 4]] | ||
== See Also == | == See Also == |
Latest revision as of 18:39, 17 August 2020
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 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 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 .