Summation Simplification involving binomial coefficient

Question

May I know what is the approach of simplifying the summation:

Since the (-1)^m term alters between -1 and 1, is that possible that this summation eventually cancels to some concise expression involving powers?

Answer 1

Remember the binomial theorem: for any real numbers $x$ and $y$, $$ (x+y)^n = \sum_{m=0}^{n} \binom{n}{m} x^m y^{n-m}. $$

You can evaluate the original sum by realizing it as an instance of the binomial theorem. First rewrite the original sum as $$ \sum_{m=0}^{n} \binom{n}{m} (-3)^m (1)^{n-m} $$ (remember that $\binom{n}{m} = \binom{n}{n-m}$). Then according to the binomial theorem, this sum is equal to $$ (-3 + 1)^n = (-2)^n. $$