Here's a way to justify the answer: is a martingale, and it is positive, hence it converges almost-surely to a random variable . We deduce, taking logarithm, that either converges to a finite limit (if ) or to (if ). Of course, since is an integer, the first case implies that this sequence is constant eventually. Obviously this can only happen if the population dies out. I let you devise your own argument for that (for instance, show for all and conclude, or just recall a result on Markov chains).

As a consequence, is either equal to 0 or 1, corresponding to a growth to infinity and to extinction respectively, and thus

.

(The martingale is bounded hence the middle equality is just a consequence of the bounded convergence theorem ; no optional sampling here or whatever)