## solving recurrences using generating functions

using the method of generating functions to solve:
$\displaystyle t_n = 3t_n-2 + 5$, with initial conditions $\displaystyle t_0 = 1$and $\displaystyle t_1 = 2$.

I have used $\displaystyle g(x) = t_0 + t(_1)x+t(_2)x^2+.....= \displaystyle\sum_{n=0}^{\infty}\$$\displaystyle t_nX^n and I have solved further and got: \displaystyle 1 + 2x + 3x^2 (\displaystyle\sum_{k=0}^{\infty}\$$\displaystyle t_kX^k)+ (\displaystyle\frac{5}{1-x}-5-5x)$

but now I'm having difficulty going about the rest. I have tried using partial fractions and such but It looks very ugly. have I made some mistake that I don't see??

Any suggestions?