Show $\sum_{k = 1}^n (k+1)2^k = n2^{n+1}$ by mathematical induction.
I made my base case $n = 1$. So $(1+1)*2^1 = 1(2^1+1)$ and $4 = 4$, which is true.
My inductive hypothesis is assuming $n = m$, the summation between $k = 1$ to $ m$ of $(k+1)*(2^k) = (m)*2^{m+1}$.
I need to show that $n = m + 1$ is true. So I did the summation $k = 1$ to $m + 1$ of $(k+1)*(2^k) = (m+1)*2^{(m+1)+1}$. But, I am unsure of how to proceed from here.
You have to show that $\sum_{k=1}^{m+1}(k+1)2^k=(m+1)2^{m+2}$, presumably using the induction hypothesis that $\sum_{k=1}^m(k+1)2^k=m2^{m+1}$. I’ll get you started:
$$\begin{align*} \sum_{k=1}^{m+1}(k+1)2^k&=\sum_{k=1}^m(k+1)2^k+\left((m+2)2^{m+1}\right)\\ &\overset{(*)}=m2^{m+1}+(m+2)2^{m+1}\\ &=\ldots \end{align*}$$
The starred step is where the induction hypothesis is used.