Show that the sequence of positive integers given by $\displaystyle a_{2n} = 3a_n -1, \ \ a_{2n+1}=3a_n +1$ is strictly increasing.

I have written down a reductio-ad-absurdum-proof, but it's based largely on tedious reduction of indices in $\displaystyle a$ and eventually showing that the argument is proved by the contradiction $\displaystyle 1\geq 2$, and now I'm not all that sure it is correct. Do you have any better ideas?