<s>[INST] If $\,x-\frac 1 x=k, \, k$ being any integer,then $\,\,x^5-\frac {1}{x^5}=?$ I am stuck with the following problem which one of friends gave me :

If $\,x-\frac 1 x=k, \, k$ being any integer,then $\,\,x^5-\frac {1}{x^5}=?$
The options are $\,\,k^5+4k^3+4k, \,k^5+5k^3+6k,\,k^5+5k^3+5k,\,k^5+5k^3+4k $.

We see that $x-\frac 1 x=k \implies x=\frac{k \pm \sqrt{k^2+4}}{2}$. Now putting this value to $\,\,x^5-\frac {1}{x^2}$ makes the calculation complicated.
Can anyone help? Thanks and regards to all.
EDIT: The problem contained a typo and thanks to @noam for pointing that out. Now using binomial expansion of $x^5-\frac{1}{x^5}$, we see that option 3 is the correct choice.
 [/INST] Let $a=x,b=\frac 1x$ and hence $$a^5-b^5=(a-b)(a^4+a^3b+a^2b^2+ab^3+b^4)$$ where $a-b=k,a^2+b^2=k^2+2,\,a^4+b^4=k^4+4k^2+2$. Now putting this values in the expression ,we get $a^5-b^5=k^5+5k^3+5k$.
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