Is there a formula for $(a+b)^n+(a-b)^n$?

Question

Is there a formula for $(a+b)^n+(a-b)^n$? Just curious. Thanks.

Answer 1

Yes there is: \begin{align*} \sum_{k=0}^n\binom{n}{k}a^{n-k}b^{k}+\sum_{k=0}^n(-1)^{k}\binom{n}{k}a^{n-k}b^{k} & = \sum_{k=0}^n \binom{n}{k}a^{n-k}b^k\left(1+(-1)^k\right) \\ & = 2\sum_{k \text{ even } \\ 0 \leq k \leq n} \binom{n}{k}a^{n-k}b^k. \end{align*}