So basically what we are asked to do is:

Marc has 1$ and plays a game in which he has 1/3 chance to gain 3 times the previous amount and 2/3 chance to divide it in 3 every round. At the end of round, he keeps the money and plays with it again. The first question is: compute E(X_1) and E(X_2) then find, by generalizing, E(X_n) where n is the number of times he plays and X_n is the amount he has after he played n times.

I found E(X_1) = 11/9 and E(X_2) = 121/81 by using LOTP, but I can't figure out how to generalize it with X_n.

If someone can help me, that would be so cool