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. |
| + | == Finite Sum == | ||
| + | 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>. | ||
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| + | '''Proof:''' | ||
| + | |||
| + | <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 <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> | ||
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| + | <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> | ||
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| + | <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> | ||
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| + | <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> | ||
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| + | <math>S_n(r-1)=(a_1+(n-1)d)g_1r^n-a_1g_1-\frac{dg_1r(r^{n-1}-1)}{r-1}</math> | ||
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| + | <math>S_n=\frac{(a_1+(n-1)d)g_1r^n}{r-1}-\frac{a_1g_1}{r-1}-\frac{dg_1r(r^{n-1}-1)}{(r-1)^2}=\frac{a_ng_{n+1}}{r-1}-\frac{x_1}{r-1}-\frac{d(g_{n+1}-g_2)}{(r-1)^2}</math> | ||
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| + | == 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 ratio 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>. | ||
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| + | <math>S=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots</math> | ||
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| + | <math>rS=a_1g_1r+(a_1g_1+dg_1)r^2+(a_1g_1+2dg_1)r^3+.\,.\,.</math> | ||
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| + | <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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| + | <math>S=\frac{dg_1r}{(r-1)^2}-\frac{a_1g_1}{r-1}=\frac{dg_2}{(r-1)^2}-\frac{x_1}{r-1}=\frac{dg_2}{(1-r)^2}+\frac{x_1}{1-r}</math> | ||
| + | |||
| + | == 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 == | ||
| + | * [[Sequence]] | ||
| + | * [[Arithmetic sequence]] | ||
| + | * [[Geometric sequence]] | ||
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.
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 .
