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I recently saw the the term $i^{x+2} = i^{x+1} + i^x$ but I can't imagine why it's true. There has to be some simple proof for it. Can you help me out? (A hint would be enough :) )

  • 2
    Is it true for $x = 0$? How about when $x = 1$? – 2017-01-12
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    If it's true for some $x$, what happens to the equation if you divide by $i^x$? – 2017-01-12
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    @quasi Then it doesn't seem true anymore... Thanks – 2017-01-12

2 Answers 2

1

Let

$i^{x+2} - i^{x+1}=i^{x} $

$i^{x+1} (i-1)=i^x$

$i^{1} (i-1)=1$

$i^2 - i =1$

$-1-i=1$

$-i\neq 2$

So your equation do not hold for any value of $x$.

1

This is not true.

The RHS of the equation i.e. $i^{x+1}+i^x$ has both a non-zero real part and an imaginary part since for all real $x$, the expression will contain a term with the imaginary unit $i$.

But the LHS of the equation i.e. $i^{x+2}$ has only a real part or an imaginary part but not both.

So the only possible case is that the equation has no solution.