Mathematical Induction Proof - Exponent with n in denominator Use mathematical induction to prove the following:
$$\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{n}{2^n}=2-\frac{n+2}{2^n}; n ∈ N $$ 
I am having trouble figuring out how to solve this with an exponent in the denominator.
$$2-\frac{n+2}{2^n}+\frac{n+1}{2^{n+1}} $$

$$\begin{align*}\underbrace{\left(\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+...+\frac{n}{2^n}\right)}_{=2-\frac{n+2}{2^n}}+\frac{n+1}{2^{n+1}}&=\left(2-\frac{n+2}{2^n}\right)+\frac{n+1}{2^{n+1}}\\&=2-\left(\frac{2(n+2)}{2^{n+1}}-\frac{n+1}{2^{n+1}}\right)\\\\&=2-\frac{(n+1)+2}{2^{n+1}}\end{align*}$$
