smallest value of $k$ for which $a_k$ is $0$
Let $b_n = a_{n+1}a_n$. Then $b_1 = 200$ and $b_{n+1} = b_n - 4$. It follows that $k=51$ is the smallest value for which $b_k = 0$ and therefore $k=52$ is the smallest value for which $a_k = 0$.
Let $b_n = a_{n+1}a_n$. Then $b_1 = 200$ and $b_{n+1} = b_n - 4$. It follows that $k=51$ is the smallest value for which $b_k = 0$ and therefore $k=52$ is the smallest value for which $a_k = 0$.