Proving $\sum_{k=0}^n \binom n k (-2)^n =(-1)^n$

Question

Can someone prove that $$\sum_{k=0}^n \binom n k (-2)^k =(-1)^n $$ ?

Answer 1

Since $$(1+x)^n = \sum_{k=0}^n \binom{n}{k} x^k$$ then $x=-2$ gives $$(1-2)^n = \sum_{k=0}^n \binom{n}{k} (-2)^k = (-1)^n$$

Answer 2

Apply the binomial theorem to $(1-2)^n$.