Count elapsed updates separately from the week label

An insect colony initially has $50$ insects. Each week its population doubles, then $30$ die through competition and limited resources. Let $u_n$ be the population at the beginning of week $n$, $n=1,2,\ldots$, with all remaining insects alive.

(i) Form a recurrence relating week n and week n−1.1 mark

(ii) Find the minimum number of weeks needed for the population to exceed $300000$.2 marks