Proof by induction that $3^n - 1$ is an even number How to demonstrate that  $3^n - 1$ is an even number using the principle of induction? 
I tried taking that $3^k - 1$ is an even number and as a thesis I must demonstrate that $3^{k+1} - 1$ is an even number, but I can't make a logical argument. 
So I think it's wrong the assumption... 
Thanks

First,  $ 3^0 - 1 = 0 $ which is even so $3^n - 1 $ is even for $ n = 0 $
Suppose that 
$3^k - 1 $ is even. We need to show that $3^{k+1}- 1 $ is even. Well, 
$ 3^{k+1}-1 = 3(3^{k}) -1 = 3(3^k - 1) + 2 $  
By our assumption, $ 3^k-1 $ is even so $ 3^{k} - 1 = 2m $, for some integer $m$
Thus, $ 3(3^k - 1) + 2 = 3(2m) + 2 = 2(3m + 1) $ Note that, $3m + 1 \in Z $ since addition and multiplication are closed in $Z$. Thus, $2(3m + 1) = 3^{k+1} -1 $ is even which is want we need to show.
Therefore, $3^n -1$ is even for all $n > 0$.
