Using mathematical induction to solve a problem
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I was solving some mathematical induction problems when I ran into this one and couldn't figure out how to solve it.
Would someone be able to give me a breakdown on how to complete it?
$$frac23+frac29+frac2{27}+...+frac{2}{3^n}=1-frac1{3^n}$$
discrete-mathematics induction
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add a comment |
$begingroup$
I was solving some mathematical induction problems when I ran into this one and couldn't figure out how to solve it.
Would someone be able to give me a breakdown on how to complete it?
$$frac23+frac29+frac2{27}+...+frac{2}{3^n}=1-frac1{3^n}$$
discrete-mathematics induction
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As an aside, note that this is true for the same reason as $0.999 = 1-frac{1}{1000}$.
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– Patrick Stevens
Dec 18 '18 at 21:41
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The sum on left is a (finite) geometric series, ratio $1/3.$
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– coffeemath
Dec 18 '18 at 21:45
add a comment |
$begingroup$
I was solving some mathematical induction problems when I ran into this one and couldn't figure out how to solve it.
Would someone be able to give me a breakdown on how to complete it?
$$frac23+frac29+frac2{27}+...+frac{2}{3^n}=1-frac1{3^n}$$
discrete-mathematics induction
$endgroup$
I was solving some mathematical induction problems when I ran into this one and couldn't figure out how to solve it.
Would someone be able to give me a breakdown on how to complete it?
$$frac23+frac29+frac2{27}+...+frac{2}{3^n}=1-frac1{3^n}$$
discrete-mathematics induction
discrete-mathematics induction
asked Dec 18 '18 at 21:35
bsoleimanybsoleimany
81
81
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As an aside, note that this is true for the same reason as $0.999 = 1-frac{1}{1000}$.
$endgroup$
– Patrick Stevens
Dec 18 '18 at 21:41
$begingroup$
The sum on left is a (finite) geometric series, ratio $1/3.$
$endgroup$
– coffeemath
Dec 18 '18 at 21:45
add a comment |
$begingroup$
As an aside, note that this is true for the same reason as $0.999 = 1-frac{1}{1000}$.
$endgroup$
– Patrick Stevens
Dec 18 '18 at 21:41
$begingroup$
The sum on left is a (finite) geometric series, ratio $1/3.$
$endgroup$
– coffeemath
Dec 18 '18 at 21:45
$begingroup$
As an aside, note that this is true for the same reason as $0.999 = 1-frac{1}{1000}$.
$endgroup$
– Patrick Stevens
Dec 18 '18 at 21:41
$begingroup$
As an aside, note that this is true for the same reason as $0.999 = 1-frac{1}{1000}$.
$endgroup$
– Patrick Stevens
Dec 18 '18 at 21:41
$begingroup$
The sum on left is a (finite) geometric series, ratio $1/3.$
$endgroup$
– coffeemath
Dec 18 '18 at 21:45
$begingroup$
The sum on left is a (finite) geometric series, ratio $1/3.$
$endgroup$
– coffeemath
Dec 18 '18 at 21:45
add a comment |
2 Answers
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I will give you the induction step $nmapsto n+1$:
$frac{2}{3}+ldots + frac{2}{3^{n+1}} = left(1-frac{1}{3^n}right)+frac{2}{3^{n+1}} = 1 + left(frac{2}{3^{n+1}}-frac{1}{3^n}right) = 1+frac{2}{3^{n+1}}-frac{3}{3cdot 3^{n}}= 1-frac{1}{3^{n+1}}$
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add a comment |
$begingroup$
Since you specifically ask about induction:
For n= 1 this is $frac{2}{3}= frac{1}{1- frac{1}{3}}$ which is true.
Now, assume that, for some k, $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}= 1- frac{1}{3^k}$. Then with $n= k+ 1$ we have $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}+ frac{2}{3^{k+1}}= 1- frac{1}{3^k}+ frac{2}{3^{k+1}}= 1- frac{3}{3^{k+1}}+ frac{2}{3^{k+1}}= 1- frac{1}{3^{k+1}}$.
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Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
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active
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active
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votes
$begingroup$
I will give you the induction step $nmapsto n+1$:
$frac{2}{3}+ldots + frac{2}{3^{n+1}} = left(1-frac{1}{3^n}right)+frac{2}{3^{n+1}} = 1 + left(frac{2}{3^{n+1}}-frac{1}{3^n}right) = 1+frac{2}{3^{n+1}}-frac{3}{3cdot 3^{n}}= 1-frac{1}{3^{n+1}}$
$endgroup$
add a comment |
$begingroup$
I will give you the induction step $nmapsto n+1$:
$frac{2}{3}+ldots + frac{2}{3^{n+1}} = left(1-frac{1}{3^n}right)+frac{2}{3^{n+1}} = 1 + left(frac{2}{3^{n+1}}-frac{1}{3^n}right) = 1+frac{2}{3^{n+1}}-frac{3}{3cdot 3^{n}}= 1-frac{1}{3^{n+1}}$
$endgroup$
add a comment |
$begingroup$
I will give you the induction step $nmapsto n+1$:
$frac{2}{3}+ldots + frac{2}{3^{n+1}} = left(1-frac{1}{3^n}right)+frac{2}{3^{n+1}} = 1 + left(frac{2}{3^{n+1}}-frac{1}{3^n}right) = 1+frac{2}{3^{n+1}}-frac{3}{3cdot 3^{n}}= 1-frac{1}{3^{n+1}}$
$endgroup$
I will give you the induction step $nmapsto n+1$:
$frac{2}{3}+ldots + frac{2}{3^{n+1}} = left(1-frac{1}{3^n}right)+frac{2}{3^{n+1}} = 1 + left(frac{2}{3^{n+1}}-frac{1}{3^n}right) = 1+frac{2}{3^{n+1}}-frac{3}{3cdot 3^{n}}= 1-frac{1}{3^{n+1}}$
answered Dec 18 '18 at 21:50
Card_TrickCard_Trick
935
935
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$begingroup$
Since you specifically ask about induction:
For n= 1 this is $frac{2}{3}= frac{1}{1- frac{1}{3}}$ which is true.
Now, assume that, for some k, $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}= 1- frac{1}{3^k}$. Then with $n= k+ 1$ we have $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}+ frac{2}{3^{k+1}}= 1- frac{1}{3^k}+ frac{2}{3^{k+1}}= 1- frac{3}{3^{k+1}}+ frac{2}{3^{k+1}}= 1- frac{1}{3^{k+1}}$.
$endgroup$
add a comment |
$begingroup$
Since you specifically ask about induction:
For n= 1 this is $frac{2}{3}= frac{1}{1- frac{1}{3}}$ which is true.
Now, assume that, for some k, $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}= 1- frac{1}{3^k}$. Then with $n= k+ 1$ we have $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}+ frac{2}{3^{k+1}}= 1- frac{1}{3^k}+ frac{2}{3^{k+1}}= 1- frac{3}{3^{k+1}}+ frac{2}{3^{k+1}}= 1- frac{1}{3^{k+1}}$.
$endgroup$
add a comment |
$begingroup$
Since you specifically ask about induction:
For n= 1 this is $frac{2}{3}= frac{1}{1- frac{1}{3}}$ which is true.
Now, assume that, for some k, $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}= 1- frac{1}{3^k}$. Then with $n= k+ 1$ we have $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}+ frac{2}{3^{k+1}}= 1- frac{1}{3^k}+ frac{2}{3^{k+1}}= 1- frac{3}{3^{k+1}}+ frac{2}{3^{k+1}}= 1- frac{1}{3^{k+1}}$.
$endgroup$
Since you specifically ask about induction:
For n= 1 this is $frac{2}{3}= frac{1}{1- frac{1}{3}}$ which is true.
Now, assume that, for some k, $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}= 1- frac{1}{3^k}$. Then with $n= k+ 1$ we have $frac{2}{3}+ frac{2}{9}+ cdotcdotcdot+ frac{2}{3^k}+ frac{2}{3^{k+1}}= 1- frac{1}{3^k}+ frac{2}{3^{k+1}}= 1- frac{3}{3^{k+1}}+ frac{2}{3^{k+1}}= 1- frac{1}{3^{k+1}}$.
answered Dec 18 '18 at 22:02
user247327user247327
11.1k1515
11.1k1515
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$begingroup$
As an aside, note that this is true for the same reason as $0.999 = 1-frac{1}{1000}$.
$endgroup$
– Patrick Stevens
Dec 18 '18 at 21:41
$begingroup$
The sum on left is a (finite) geometric series, ratio $1/3.$
$endgroup$
– coffeemath
Dec 18 '18 at 21:45