How to prove this is a rational number I'm not sure how to prove this is a rational number $\frac{q}{m}$, can some one show me?
$$\frac{q}{m}=\frac{(\frac{1+\sqrt5}{2})^n - (\frac{1-\sqrt5}{2})^n}{\sqrt5}$$

Lt me consider $$S(n) = 2\frac{ \left((a+b)^n-(a-b)^n\right)}{b}$$The binomial theorem will be applied to each term and the result will be expanded and simplified. So, the results are successively $$S(1) = 4$$ $$S(2) = 8 a$$  $$S(3)= 4\left(3 a^2+b^2\right)$$ $$S(4)= 16 a \left(a^2+b^2\right)$$ $$S(5)=4 \left(5 a^4+10 a^2 b^2+b^4\right)$$  $$S(6)=8 a \left(3 a^2+b^2\right) \left(a^2+3 b^2\right)$$ $$S(7)=4 \left(7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right)$$ $$S(8)=32 a \left(a^2+b^2\right) \left(a^4+6 a^2 b^2+b^4\right)$$  
So, you see, as clearly pointed out by the previous comments and answers, that if $a$ is rational and $b$ the square root of a rational, the result is rational. 
