MathNet 2013 Question 34

a sequence $a_0, a_1, \dots$ is defined by $a_0 = 1$ and for $k \ge 1$, $a_{2k} = (-1)^k a_k$; for $k \ge 0$, $a_{2k+1} = -a_k$. Prove that for every $n \ge 0$, $a_0 + a_1 + \dots + a_n \ge 0$.