Recursive Definition of a Series

Question

I have a series such as the one below:

\begin{equation} 2^n(\sum\limits_{i=2}^{n+1}i)\text{ for all $n \geq 1$} \end{equation}

I need to write a recursive definition for it. Here's what I have so far:

\begin{equation} sum(n)=\begin{cases} n, & \text{if $n<1$}.\\ recurse, & \text{otherwise}. \end{cases} \end{equation}

I'm not entirely sure what the recursive step would be or if the return for a value less than 1 should be n. Can anybody help me out? Thanks!

Answer 1

$$\begin{align} sum(0)&=0\\ sum(n\geq1)&=2^n\sum_{i=2}^{n+1}i \\ &=2\cdot\left(2^{n-1}\sum_{i=2}^{n}i\right)+2^n\cdot(n+1)\\ &=2\cdot sum(n-1)+2^n\cdot(n+1) \end{align}$$

Or, \begin{equation} sum(n)=\begin{cases} n, & \text{if $n<1$}.\\ 2\cdot sum(n-1)+2^n\cdot(n+1), & \text{otherwise}. \end{cases} \end{equation}