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

I have used g(x) = t_0 + t(_1)x+t(_2)x^2+.....= \displaystyle\sum_{n=0}^{\infty}\  t_nX^n

and I have solved further and got:

1 + 2x + 3x^2 (\displaystyle\sum_{k=0}^{\infty}\  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?