Use the binomial theorem to derive a closed form expression for ${n \choose 0} + {n \choose 4} + {n \choose 8} +...+{n \choose 4⌊n/4⌋}$

Question

Use the binomial theorem to derive a closed form expression for $${n \choose 0} + {n \choose 4} + {n \choose 8} +...+{n \choose 4⌊n/4⌋}$$ And I should use imaginary numbers right?

Answer 1

Hint: Start with $(1+x)^n = \displaystyle\sum_{k = 0}^{n}\dbinom{n}{k}x^k$, and plug in $x = 1,i,-1,-i$. This will give you four equations. What do you get when you add these four equations together and divide by $4$?

Also see this related question.