Q. Solve the following recurrence relation with the initial conditions: $a_n=a_{n-1}+a_{n-2}$ for $n\ge2,$ and $a_0=0,a_1=1.$
I started with $G(x)=\displaystyle\sum_{n=0}^{\infty}a_nx^n,$ and after using the given conditions obtained:
$G(x)=\dfrac x{1-x-x^2}.\tag*{}$
I'm not sure how to proceed further with it so that I can find $a_n$ explicitly. Using binomial theorem for negative exponents (namely $-1$) also won't work here as there will be an additional factor of the form $(1-x)^n$ alongside $x^n$, where $n=0,1,2,\ldots$ . So please suggest me a way out. Thanks in advance..