Finding mean and variance of a population problem

Question

A population beings with a single individual. In each generation, each individual in the population dies with probability $1/2$ or doubles with probability $1/2$. If I let $X_n$ denote the number of individuals in the population in the $nth$ generation, how would I find the mean and variance?

So since the mean population size decreases geometrically when $\mu < 1$, wouldn't the mean be $\mu^{-n}$? To find variance would I just square the mean and use the variance formula then?

Answer 1

I think I'm calculating this wrong or guessing wrong if calculations right but I think $E[X_n] = 1$ and $Var[X_n] = n-1$?

Reason for guess (not full proof if ever correct) is that I think:

$X_1 = 1$;

$X_2 = 0$, w/p=1/2,

$X_2 = 2$, w/p=1/2;

$X_3 = 4$, w/p=1/8,

$X_3 = 2$, w/p=2/8,

$X_3 = 0$, w/p=5/8;

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