Question

inability versus Growth. Suppose young entrepreneurs running a lemonade stand make a $10 nday one (i = 1) and then take half of the profit to reinvest in preparation for the next day. n day two, they make $5 and reinvest $10/3, and so on for subeequent days on to infinity. Compose 4rofit on day one (i 1 tbe sum of the net daily balance over all days and determine its total value.

Answer 1

On Day 1: Profit = 10, reinvestment = 1/2 (10) =5. Balance = 10- 10/2 = 5
Day 2: Profit = 5, reinvestment = 1/3(10) = 3.33, Balance= 10/2 - 10/3

On nth Day, Profit = 10/n and reinvestment = 10 * 1/(n+1), Balance = 10/n - 10/(n+1)
On summing this series to infinity: Left term for each day cancels with the right term for prev day, leaving behind 10- 10/(n+1).

For n going to infinity, 1/(n+1) goes to Zero, and hence we get the sum of the series as 10!

Let me know if you have any queries with this approach.