A number theory problem by Wildan Bagus Wicaksono

Number Theory Level 1

$1+\frac { 1 }{ n } +\frac { 1 }{ { n }^{ 2 } } +\frac { 1 }{ { n }^{ 3 } } +\cdots=\frac { x }{ y }$

Let $x,y$ and $n$ be positive integers larger than 1.

Given that $x$ and $y$ are coprime. Find $x-y$ .

The answer is 1.

1 solution

Marta Reece
Jul 2, 2017

$1+\frac { 1 }{ n } +\frac { 1 }{ { n }^{ 2 } } +\frac { 1 }{ { n }^{ 3 } } +...$ is a geometric series with first term $a=1$ and common ratio $r=\frac 1n$ .

The sum of the series is $S=\dfrac a{1-r}=\dfrac 1{1-\frac 1n}=\dfrac n{n-1}=\dfrac xy$

The difference $x-y=n-(n-1)=\boxed1$

A number theory problem by Wildan Bagus Wicaksono

The answer is 1.

1 solution

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