Using mathematical induction to solve a problem

Question

I was solving some mathematical induction problems when I ran into this one and couldn't figure out how to solve it.

Would someone be able to give me a breakdown on how to complete it?

$$\frac23+\frac29+\frac2{27}+...+\frac{2}{3^n}=1-\frac1{3^n}$$

Answer 1

I will give you the induction step $n\mapsto n+1$:

$\frac{2}{3}+\ldots + \frac{2}{3^{n+1}} = \left(1-\frac{1}{3^n}\right)+\frac{2}{3^{n+1}} = 1 + \left(\frac{2}{3^{n+1}}-\frac{1}{3^n}\right) = 1+\frac{2}{3^{n+1}}-\frac{3}{3\cdot 3^{n}}= 1-\frac{1}{3^{n+1}}$

Answer 2

Since you specifically ask about induction: For n= 1 this is $\frac{2}{3}= \frac{1}{1- \frac{1}{3}}$ which is true.

Now, assume that, for some k, $\frac{2}{3}+ \frac{2}{9}+ \cdot\cdot\cdot+ \frac{2}{3^k}= 1- \frac{1}{3^k}$. Then with $n= k+ 1$ we have $\frac{2}{3}+ \frac{2}{9}+ \cdot\cdot\cdot+ \frac{2}{3^k}+ \frac{2}{3^{k+1}}= 1- \frac{1}{3^k}+ \frac{2}{3^{k+1}}= 1- \frac{3}{3^{k+1}}+ \frac{2}{3^{k+1}}= 1- \frac{1}{3^{k+1}}$.