Question 155
Let a sequence of real numbers $a_0, a_1, a_2, \dots$ satisfies the condition
$$
\sum_{n=0}^{m} a_n \cdot (-1)^n \cdot \binom{m}{n} = 0
$$
for all sufficiently large values of $m$. Show that there exists a polynomial $P$ such that $a_n = P(n)$ for all $n \ge 0$.
$$
\sum_{n=0}^{m} a_n \cdot (-1)^n \cdot \binom{m}{n} = 0
$$
for all sufficiently large values of $m$. Show that there exists a polynomial $P$ such that $a_n = P(n)$ for all $n \ge 0$.