sequences and series - Prove by induction that $sum_{n=1}^infty frac{1}{2^n} = 1$

The title explains the problem fairly well; is there a way to prove by induction that $\sum_{n=1}^\infty \frac{1}{2^n} = 1$. If not are there other ways?

I have thought of showing it by rewriting the series so that.

$$\sum_{n=1}^\infty \frac{1}{2^n} = 1 \implies \sum_{n=1}^\infty \frac{1}{2}(\frac{1}{2})^{n-1} = 1$$

And then from there conclude that it is a geometric series with the values $r = 1/2$ and $a=1/2$ thus

$$\sum_{n=1}^\infty \frac{1}{2^n} = \frac{1/2}{1-1/2} = 1$$

This seems like kind of a vodoo proof, so i was wondering if its possible to do this by induction?