Twist on St. Petersberg Paradox

Question

Consider the St. Petersburg paradox, except that you receive $n$ rather than 2ⁿ if the game lasts for n rounds. What is the fair value of this game? What if the payoff is $n$²?

So, after doing the St. Petersburg paradox, I calculated the E(2ⁿ) = 4. If we simply make it n, would it not just be 2? And $n$² is 16? Or is that mistaken?

Thanks!