Proof needed for the combinatorial identity $\sum_{a=0}^n\binom{n+a}{a}/{2^{n+a}}=1$

Question

I need some algebraic and combinatorial proofs for the following.

$$\sum_{a=0}^n\frac{\binom{n+a}{a}}{{2^{n+a}}}=1.$$

Every kind of using combinatorial consideration, generating function, algebraic simplification, reduction to some famous formula, etc is appreciated.

Thanks in advance.