Let $F_n$ be the $n$-th Fibonacci number ($F_1$ = $F_2$ = $1$ and $F_{n+1}$ = $F_n$ + $F_{n-1}$ for $n \geq 2$). Let $P_n(x)$ be a polynomial of degree $n$ such that $P_n(k)$ = $F_k$ for $k = n+2, n+3,\dots,2n+2$. Prove that $P(2n+3)$ = $F_{2n+3} -1$.
I have attempted it through straightforward application of Lagrange interpolation method. I have arrived at a crazy summation. I have no idea if any of what I have done is right. If you could provide some details for the solution that would also help me realise if I'm dabbling out of my depth here and should start with simpler lessons.