How to do this Math induction problem? Show that: 
$$\frac n3 + \frac n9 + \frac {n}{27} + \cdots = \frac n2.$$ 
When I start with $\frac 13 + \frac 19 + \frac {1}{27}$ it leads to a number close to $.5$ but it's not exactly $.5$.

You have an infinite geometric series $\frac n3 + \frac n9 + \frac n{27}+\dots=n\sum_{i=1}^\infty\frac 1{3^n}$.  Using the terminology in the linked article, we have $a=\frac 13$ (the first term) and $r=\frac 13$ (the ratio between successive terms.  As long as $|r| \lt 1$ this infinite sum will converge to $\frac a{1-r}=\frac {\frac 13}{1-\frac 13}=\frac 12$
