Bacteria branching process

Question

The population starts with one individual. Per one period, the individual can either split into two, survive, or die, each with probability one third. What is the expected population size after n periods?

Answer 1

Hint: Define $Y_n$ the amount of individuals at step $n$, and notice that if $Y_n=m$ then you have $m$ R.V-s $X_{i}$, where each $X_i$ satisfies

$$ \mathbb{P}(X_i=1),\mathbb{P}(X_i=2),\mathbb{P}(X_i=-1)=\frac{1}{3}. $$

Note that $\sum_{i=1}^{Y_n} X_i=Y_{n+1}$ and that this gives you a simple identity for $\mathbb{E}[Y_{n+1}\vert Y_n=m].$ Can you carry on from this point?