Answer
$a_{n}=\displaystyle \frac{1+(-1)^{n+1}}{2}, \quad n=1,2,3...$
Work Step by Step
Using: $\quad (-1)^{n}=\left\{\begin{array}{ll}
+1, & \text{ when n is even}\\
-1, & \text{ when n is odd}
\end{array}\right.$
We see that
$1+(-1)^{n}=\left\{\begin{array}{ll}
2, & \text{ when n is even}\\
0, & \text{ when n is odd}
\end{array}\right.$
We want odd terms to give nonzero values
$1+(-1)^{n+1}=\left\{\begin{array}{ll}
2, & \text{ when n is odd}\\
0, & \text{ when n is even}
\end{array}\right.$
So, a general term $a_{n}=1+(-1)^{n+1}$
generates$\quad 2,0,2,0,2,...$
We obtain the problem sequence by dividing each term of this sequence by two:
$a_{n}=\displaystyle \frac{1+(-1)^{n+1}}{2}, \quad n=1,2,3...$
