discrete mathematics - Proof by contradiction and mathematical induction

$\sum_{i=1}^n {2\over3^i}={2\over3}+{2\over9}+\dots+{2\over3^n}=1-{({1\over3})^n}$

I had this problem in class and we proved using 2 different methods: contradiction and mathematical induction. I thought it was understood, I just got bumped into certain point.

Please point it out which step I'm thinking wrong.

For the contradiction,

We assume that there is some integer n for which $i=1$ is false.
And we are applying smaller positive integer smaller than 1.

for the smallest n, ${2\over3}+{2\over9}+{\dots}+{1\over3^{n-1}}$ indicates that our assumption $i=1$ is false.

(I don't remember how the calculation was made for this proof by contradiction.)

Therefore, our assumption was true.

For induction,

Try out the base case with applying $i=1$
inductive hypothesis would be ${2\over3}=1-{1\over3}$

What would be the next step?