Induction proof: $\sum_{k=1}^n k^32^k \leq n^32^{n+1}$

Question

Having trouble solving the induction proof for:

$$\sum_{k=1}^n k^32^k \leq n^32^{n+1}$$

EDIT: supposed to be $k^32^k$

Answer 1

Hint: $$\sum_{k=1}^{n+1}k^3 2^k = \sum_{k=1}^nk^32^k + (n+1)^3 2^{n+1} \leq n^32^{n+1} + (n+1)^3 2^{n+1} = \dots$$

Answer 2

For $n=1$ the claim is clear.

Suppose $\sum_{k=1}^nk^22^k\leq n^32^{n+1}$. Now

\begin{align} \sum_{k=1}^{n+1}k^22^k&=\sum_{k=1}^nk^22^k+(n+1)^22^{n+1}\\&\leq n^32^{n+1}+(n+1)^22^{n+1}\\ &=(n^3+(n+1)^2)2^{n+1}\\ &=(n^3+n^2+2n+1)2^{n+1}\\ &\leq(n^3+3n^2+3n+1)2^{n+2}\\ &=(n+1)^32^{(n+1)+1} \end{align}

And the proof is complete.