A number theory problem by Akash Shukla

Assume that $a_{n} = 0$ for some integer n.
$a_{n} = a_{n-1} - a_{n-2}$
$a_{n} = 0 \to a_{n-1} = a_{n-2}$
$a_{n-1}= a_{n-2} = a_{n-2} - a_{n-3} \to a_{n-3} = 0$
Case 1:
$n$ is even.
$a_{n} = 0 \to a_{n-3} = 0 \to a_{n-6} = 0 \ldots a_{1} =0$ which is true.
Case 2:
$n$ is odd.
$a_{n} = 0 \to a_{n-3} = 0 \ldots \to a_{0} = 0$ which is true as well.
$\therefore a_{n} = 0, n \in Z^{+}$
$\therefore \displaystyle \sum_{n=1}^{\infty} a_{n} = 0$