A rumour spreads by each person telling two others; a savings account adds a fixed £50 a month. Why is the comparison never close in the long run, whatever the starting amounts?

A. Because the rumour's increment grows with the total and savings' does not

One step is constant and the other is a proportion of a growing total, so the second compounds and the first does not. Start the savings at £1,000 a month and the doubling rumour still passes it inside a month, which is the arithmetic behind compound interest, epidemics and why O(2ⁿ) is hopeless.