Prove by mathematical induction for all n in N

Question

Prove by mathematical induction that $$ 1+\frac12+\frac14+\frac18+\dotsb+\frac{1}{2^i} = 2 - \frac{1}{2^i} $$

I know the base set just stuck in the calculations for the inductive set.

Answer 1

Hint

$$-\frac1{2^n}+\frac1{2^{n+1}}=\frac{-2+1}{2^{n+1}}$$

Answer 2

$$1+\frac 12+\dots+\frac 1{2^i}+\frac 1{2^{i+1}}=2-\frac1{2^i}+\frac1{2^{i+1}}$$ and $\dfrac1{2^{i+1}}$ is just half $\dfrac1{2^i}$.

Answer 3

Base case: $ 1/2^0 = 2 = 2 - 1/2^0 $ is correct.

Now, assuming $ \sum^i_{k=0} \frac 1{2^k}= 2-\frac 1{2^i} $

$$ \sum^{i+1}_{k=0}\frac 1{2^k} = \sum^i_{k=0}\frac 1{2^k}+\frac 1{2^{i+1}} = 2-\frac 1{2^i}+\frac 1{2^{i+1}} = 2-\frac 1{2^{i+1}} $$