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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>.
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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)+\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+(a_1+dg_1)r+(a_1g_1+2dg_1)r^2+\ldots+(a_1g_1+(n-1)dg_1)r^{n-1}</math>
+
<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+(a_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=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>
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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>
 
<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>
 +
 +
== 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>.
 +
 +
<math>S=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots</math>
 +
 +
<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>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 ==
 
== 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: $x_n=a_ng_n$, where $a_n$ and $g_n$ are the $n$th terms of arithmetic and geometric sequences, respectively.

Finite Sum

The sum of the first $n$ terms of an $\textbf{arithmetico-geometric sequence}$ is $\frac{a_ng_{n+1}}{r-1}-\frac{x_1}{r-1}-\frac{d(g_{n+1}-g_2)}{(r-1)^2}$, where $d$ is the common difference of $a_n$ and $r$ is the common ratio of $g_n$. Or, $\frac{a_ng_{n+1}-x_1-drS_g}{r-1}$, where $S_g$ is the sum of the first $n$ terms of $g_n$.

Proof:

$x_n=(a_1+d(n-1))(g_1\cdot r^{n-1})$

Let $S_n$ represent the sum of the first $n$ terms. $S_n=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots+(a_1+(n-1)d)(g_1r^{n-1})$

$S_n=a_1g_1+(a_1g_1+dg_1)r+(a_1g_1+2dg_1)r^2+\ldots+(a_1g_1+(n-1)dg_1)r^{n-1}$

$rS_n=a_1g_1r+(a_1g_1+dg_1)r^2+(a_1g_1+2dg_1)r^3+\ldots+(a_1g_1+(n-1)dg_1)r^{n}$

$rS_n-S_n=-a_1g_1-dg_1r-dg_1r^2-dg_1r^3-\ldots-dg_1r^{n-1}+(a_1g_1+(n-1)dg_1)r^n$

$S_n(r-1)=(a_1+(n-1)d)g_1r^n-a_1g_1-\frac{dg_1r(r^{n-1}-1)}{r-1}$

$S_n=\frac{(a_1+(n-1)d)g_1r^n}{r-1}-\frac{a_1g_1}{r-1}-\frac{dg_1r(r^{n-1}-1)}{(r-1)^2}=\frac{a_ng_{n+1}}{r-1}-\frac{x_1}{r-1}-\frac{d(g_{n+1}-g_2)}{(r-1)^2}$

Infinite Sum

The sum of an infinite arithmetico-geometric sequence is $\frac{dg_2}{(1-r)^2}+\frac{x_1}{1-r}$, where $d$ is the common difference of $a_n$ and $r$ is the common ratio of $g_n$ ($|r|<1$). Or, $\frac{drS_g+x_1}{1-r}$, where $S_g$ is the infinite sum of the $g_n$.

$S=a_1g_1+(a_1+d)(g_1r)+(a_1+2d)(g_1r^2)+\ldots$

$rS=a_1g_1r+(a_1g_1+dg_1)r^2+(a_1g_1+2dg_1)r^3+.\,.\,.$

$rS-S=-a_1g_1-dg_1r-dg_1r^2-dg_1r^3-\ldots=-a_1g_1+\frac{dg_1r}{r-1}$

$S=\frac{dg_1r}{(r-1)^2}-\frac{a_1g_1}{r-1}=\frac{dg_2}{(r-1)^2}-\frac{x_1}{r-1}=\frac{dg_2}{(1-r)^2}+\frac{x_1}{1-r}$

Example Problems

See Also

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