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Proof by Induction - New to proofs

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2

$begingroup$

totally new to proofs and found this challenge problem and struggling a bit. Any help would be appreciated!

There are some real numbers $x$ such that $x+frac{1}{x}$ is an
integer. For example, $2+sqrt{3}+frac{1}{2+sqrt{3}}=4$,
$1+frac{1}{1}=2$, and $2sqrt{6}-5+frac{1}{2sqrt{6}-5}=-10$.

Prove for all $xinmathbb{R}$ that if $x+frac{1}{x}$ is an integer,
then $x^n +frac{1}{x^n}$ also is an integer for all $ninmathbb{N}$.

induction

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asked 1 hour ago

RobinRobin

261

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• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I recommend this answer as a good start
  $endgroup$
  – Ross Millikan
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  math.stackexchange.com/questions/936479/…
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  – lab bhattacharjee
  1 hour ago
  

  
  
  
  

add a comment | 

2

$begingroup$

totally new to proofs and found this challenge problem and struggling a bit. Any help would be appreciated!

There are some real numbers $x$ such that $x+frac{1}{x}$ is an
integer. For example, $2+sqrt{3}+frac{1}{2+sqrt{3}}=4$,
$1+frac{1}{1}=2$, and $2sqrt{6}-5+frac{1}{2sqrt{6}-5}=-10$.

Prove for all $xinmathbb{R}$ that if $x+frac{1}{x}$ is an integer,
then $x^n +frac{1}{x^n}$ also is an integer for all $ninmathbb{N}$.

induction

share|cite|improve this question

asked 1 hour ago

RobinRobin

261

New contributor

Robin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I recommend this answer as a good start
  $endgroup$
  – Ross Millikan
  1 hour ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  math.stackexchange.com/questions/936479/…
  $endgroup$
  – lab bhattacharjee
  1 hour ago
  

  
  
  
  

add a comment | 

$begingroup$

totally new to proofs and found this challenge problem and struggling a bit. Any help would be appreciated!

There are some real numbers $x$ such that $x+frac{1}{x}$ is an
integer. For example, $2+sqrt{3}+frac{1}{2+sqrt{3}}=4$,
$1+frac{1}{1}=2$, and $2sqrt{6}-5+frac{1}{2sqrt{6}-5}=-10$.

Prove for all $xinmathbb{R}$ that if $x+frac{1}{x}$ is an integer,
then $x^n +frac{1}{x^n}$ also is an integer for all $ninmathbb{N}$.

induction

share|cite|improve this question

asked 1 hour ago

RobinRobin

261

New contributor

Robin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked 1 hour ago

RobinRobin

261

New contributor

Robin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked 1 hour ago

RobinRobin

261

New contributor

Robin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 1 hour ago

RobinRobin

261

New contributor

Robin is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 1 hour ago

RobinRobin

261

1 Answer
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7

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HINT: Note that for $ngeq1$ you have
$$left(x^n+frac{1}{x^n}right)left(x+frac{1}{x}right)=left(x^{n+1}+frac{1}{x^{n+1}}right)+left(x^{n-1}+frac{1}{x^{n-1}}right).$$

For more details, hover over the the block below:

The equation above can be rewritten to get
$$x^{n+1}+frac{1}{x^{n+1}}=left(x^n+frac{1}{x^n}right)left(x+frac{1}{x}right)-left(x^{n-1}+frac{1}{x^{n-1}}right).$$
If the three terms in parentheses on the right hand side are integers, then so is the left hand side. Now to use induction, all you need is that $x^n+frac{1}{x^n}$ is an integer for $n=0$ and $n=1$.

share|cite|improve this answer

answered 1 hour ago

ServaesServaes

26.5k33997

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7

$begingroup$

HINT: Note that for $ngeq1$ you have
$$left(x^n+frac{1}{x^n}right)left(x+frac{1}{x}right)=left(x^{n+1}+frac{1}{x^{n+1}}right)+left(x^{n-1}+frac{1}{x^{n-1}}right).$$

For more details, hover over the the block below:

The equation above can be rewritten to get
$$x^{n+1}+frac{1}{x^{n+1}}=left(x^n+frac{1}{x^n}right)left(x+frac{1}{x}right)-left(x^{n-1}+frac{1}{x^{n-1}}right).$$
If the three terms in parentheses on the right hand side are integers, then so is the left hand side. Now to use induction, all you need is that $x^n+frac{1}{x^n}$ is an integer for $n=0$ and $n=1$.

share|cite|improve this answer

answered 1 hour ago

ServaesServaes

26.5k33997

$endgroup$

add a comment | 

7

$begingroup$

HINT: Note that for $ngeq1$ you have
$$left(x^n+frac{1}{x^n}right)left(x+frac{1}{x}right)=left(x^{n+1}+frac{1}{x^{n+1}}right)+left(x^{n-1}+frac{1}{x^{n-1}}right).$$

For more details, hover over the the block below:

The equation above can be rewritten to get
$$x^{n+1}+frac{1}{x^{n+1}}=left(x^n+frac{1}{x^n}right)left(x+frac{1}{x}right)-left(x^{n-1}+frac{1}{x^{n-1}}right).$$
If the three terms in parentheses on the right hand side are integers, then so is the left hand side. Now to use induction, all you need is that $x^n+frac{1}{x^n}$ is an integer for $n=0$ and $n=1$.

share|cite|improve this answer

answered 1 hour ago

ServaesServaes

26.5k33997

$endgroup$

add a comment | 

$begingroup$

HINT: Note that for $ngeq1$ you have
$$left(x^n+frac{1}{x^n}right)left(x+frac{1}{x}right)=left(x^{n+1}+frac{1}{x^{n+1}}right)+left(x^{n-1}+frac{1}{x^{n-1}}right).$$

For more details, hover over the the block below:

The equation above can be rewritten to get
$$x^{n+1}+frac{1}{x^{n+1}}=left(x^n+frac{1}{x^n}right)left(x+frac{1}{x}right)-left(x^{n-1}+frac{1}{x^{n-1}}right).$$
If the three terms in parentheses on the right hand side are integers, then so is the left hand side. Now to use induction, all you need is that $x^n+frac{1}{x^n}$ is an integer for $n=0$ and $n=1$.

share|cite|improve this answer

answered 1 hour ago

ServaesServaes

26.5k33997

$endgroup$

share|cite|improve this answer

answered 1 hour ago

ServaesServaes

26.5k33997

answered 1 hour ago

ServaesServaes

26.5k33997

answered 1 hour ago

ServaesServaes

26.5k33997