Add a late fee after interest and count the charging cycles

A man enters a coma after an accident on 10 January 2013 and misses payment of his credit-card balance M dollars due on 27 January. On each monthly 27th, unpaid balances first receive $2\%$ interest and then an additional late fee L dollars.

(a) Express his balance on 1 February 2013 in terms of M and L.1 mark

(b) Show that exactly n months after the accident, on the same day of the month, the accumulated balance is $(1.02)^nM+50L[(1.02)^n-1]$.3 marks

(c) For $M=1000$, $L=55$, find the least n for which the balance first exceeds $2010.2 marks