summation - Mathematical Induction with Exponents: $1 + frac12 + frac14 + dots + frac1{2^{n}} = 2 - frac1{2^{n}}$

Prove $1 + \frac{1}{2} + \frac{1}{4} + ... + \frac{1}{2^{n}} = 2 - \frac{1}{2^{n}}$ for all positive integers $n$.

My approach was to add $\frac{1}{2^{n + 1}}$ to both sides for the induction step. However, I got lost in the algebra and could not figure out the rest of the proof. Any help would be greatly appreciated.

Thank you in advance.

Answer

Hint

if we add $\frac{1}{2^{n+1}}$ to the right side, we get

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

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

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