Mathematical Induction Proof - Exponent with n in denominator

Question

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.

$$2-\frac{n+2}{2^n}+\frac{n+1}{2^{n+1}} $$

Answer 1

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

Answer 2

Hint: Just add $\frac{n+1}{2^{n+1}}$ to both sides of your original equation. Then simplify the sum on the right, noting that $2^{n+1}=2\cdot 2^n$.

Also, your fraction with $n+3$ in the numerator appears to be incorrect.