Question 19
(ROM 1) Let $(F_n)_{n\ge1}$ be the Fibonacci sequence $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$ $(n\ge1)$, and $P(x)$ the polynomial of degree $990$ satisfying
$$P(k)=F_k,\qquad k=992,\ldots,1982.$$
Prove that $P(1983)=F_{1983}-1$.
$$P(k)=F_k,\qquad k=992,\ldots,1982.$$
Prove that $P(1983)=F_{1983}-1$.