Using the binomial expansion to solve a summation

Question

I have to evaluate a summation from k=1 to n of k3^k(nCk) by setting x equal to the appropriate values in the binomial expansion.

Answer 1

Hint: Note that $\binom{n}{k}=\frac{n}{k}\binom{n-1}{k-1}$. So our sum has shape $$3n\sum \binom{n-1}{k-1}3^{k-1}.$$ The sum $\sum \binom{n-1}{k-1}3^{k-1}$ is closely related to the binomial expansion of $(1+3)^{n-1}$.