I need to prove the following identity by induction: $$\sum_{i=0}^ni\cdot n^{i-1}=(n-1)\cdot 2^n+1$$
I have the base case where $n=1$ and all that, but I'm stuck on how to turn the inductive step into the final solution. I know I need to assume the identity holds for $n-1$, but I'm stuck on how to use that to show it holds true for n.
Your identity seems wrong to me. I took $n=5$ then the L.H.S. becomes, $\displaystyle \sum_{i=0}^5 i\cdot 5^{i-1}=0+1+10+75+500+3125=3711$ but R.H.S. becomes $(5-1)\cdot 2^5+1=2^7+1=129$ whis is absurd!
Note: However, putting the LHS on Mathematica gives $$\sum_{i=0}^ni\cdot n^{i-1}=\frac{1-n^n-n^{1+n}+n^{2+n}}{(n-1)^2}$$
Hoping for this to be true for all $n\in\mathbb{N}\setminus\{0\}$.