Using the fact that you can show for all by induction.
Using the same fact, for the variance you can derive the following recurrence:
(hopefully I have not made any mistakes in my calculations)
The solution to the recurrence (using generating functions for example) is
Note I didn't actually solve the recurrence myself, I used wolfram alpha.
so there may be an error there.
Now that you know the true expectation and variance, maybe try altering your guess to a normal with those values? Tell me if it seems more reasonable. To prove the actual distribution is normal with that expectation and variance shouldn't be too hard if its true ( I just might bite my tongue on that later .)
Whats interesting to note is that