Question

Use the Binomial Theorem to prove the following:

Answer 1

\[2^n=\sum_{k=0}^{n}(-1)^k \left(\begin{matrix}n \\ k\end{matrix}\right)3^{n-k}\]

Answer 2

Also to prove that \[3^n=\sum_{k=0}^{n}\left(\begin{matrix}n \\ k\end{matrix}\right)2^k\]