<s>[INST] Find the sum of $\binom n0 - \binom n2 +\binom n4 -\binom n6 \cdots$ Find the sum of $\binom n0 - \binom n2 +\binom n4 -\binom n6 \cdots$

Using Binomial expansion of $(1+x)^n$, 
$$ \binom n0 +\binom n1 x^1 + \binom n 2 x^2 + \cdots + \binom n nx^n $$
Substituting $x = i$,
$$(1+i)^n = \binom n0 +\binom n1 i^1 + \binom n 2 i^2 + \cdots + \binom n ni^n $$
$$(1+i)^n = \binom n0 +\binom n1 i - \binom n 2  - \binom n 3 i\cdots + \binom n ni^n $$
How to proceed further?
 [/INST] So you have $$(1+i)^{n}= \binom{n}{0} + \binom{n}{1} i + \binom{n}{2}i^2 + \cdots = \binom{n}{0}-\binom{n}{2} + \binom{n}{4} + i \cdot\left[\binom{n}{1}-\binom{n}{3} + \cdots\right]$$


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*Write $(1+i)^{n}= (\sqrt{2})^{n}\cdot \left(\frac{1}{\sqrt{2}}+i \cdot\frac{1}{\sqrt{2}}\right)^{n}$ and see if you can find the real part of this term.

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