Sum of Even Index Binomial Coefficients/Proof 1

Theorem

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

$\ds \sum_{i \mathop \in \Z} \binom n i = 2^n$

That is:

$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 + \cdots + \dbinom n n = 2^n$

as $\dbinom n i = 0$ for $i < 0$ and $i > n$.

This can be written more conveniently as:

$\dbinom n 0 + \dbinom n 1 + \dbinom n 2 + \dbinom n 3 + \dbinom n 4 + \cdots = 2^n$

$\ds \sum_{i \mathop \in \Z} \paren {-1}^i \binom n i = 0$

That is:

$\dbinom n 0 - \dbinom n 1 + \dbinom n 2 - \dbinom n 3 + \dbinom n 4 - \cdots = 0$

Adding them together, we get:

$2 \dbinom n 0 + 2 \dbinom n 2 + 2 \dbinom n 4 + \cdots = 2^n$

as the odd index coefficients cancel out.

Dividing by $2$ throughout gives us the result.

$\blacksquare$