<s>[INST] Proof $\dbinom{n}{0} - \frac{1}{2}\dbinom{n}{1} + \cdots + (-1)^n\frac{1}{2^{n-1}}\dbinom{n}{n-1}$ For all $n \ge 1$,
$$\binom{n}{0} - \frac{1}{2}\binom{n}{1} + \frac{1}{2^2}\binom{n}{2} - \frac{1}{2^3}\binom{n}{3} + \cdots + (-1)^n\frac{1}{2^{n-1}}\binom{n}{n-1} = 0,$$ if $n$ is even
                                                                                  $$\frac{1}{2^{n-1}}, \text{ if } n \text{ is odd}$$
I need to use the binomial theorem in this question.
 [/INST] Try to use the binomial formula on $(1+(-\frac{1}{2}))^n$
This will give you to the proof of the statement.   
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