IR-TST 2015 Problem 17

Prove that for each natural number $d$ , There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$ ≠ $0$ and for each sequence like $a_{1}$ , $a_{2}$ , $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$