Sum of Even Index Binomial Coefficients

Theorem
Let $n > 0$ be a (strictly) positive integer.

Then:
 * $\ds \sum_{i \mathop \ge 0} \binom n {2 i} = 2^{n - 1}$

where $\dbinom n i$ is a binomial coefficient.

That is:
 * $\dbinom n 0 + \dbinom n 2 + \dbinom n 4 + \dotsb = 2^{n - 1}$

Also see

 * Sum of Odd Index Binomial Coefficients