The timeline is going to be useful here. Different things are happening to the bank account at different times of the month, and the questions are asking about the state of the bank account at different times of the month. To start, here's what happens in January: If Kendra withdraws 75% of her money, that means 25% is left, so she has (0.25)(100) = $25 in her account at the end of January. (a) Extend the timeline. At the end of January, Kendra has $25. At the beginning of February she deposits another $100, so she now has $125 in the bank. In the middle of February she withdraws 75% of her money, leaving 25% of that $125 in the bank. At the end of February her bank account contains 0.25(125) = 31.25 dollars. At the beginning of March she deposits $100, so now her account holds 100 + 31.25 = $131.25. (b) Let *M*_{n} be the amount Kendra has in the bank after her *n*th deposit. Her *n*th deposit is $100, and she still has 25% of *M*_{n – 1} left over from the previous month. This means *M*_{n} = 100 + 0.25*M*_{n – 1}.
It's not practical to use this formula for direct calculations. If we wanted to calculate, say, *M*_{20}, we would first have to calculate *M*_{1} through *M*_{19}. However, since this formula tells us how to get one value of *M*_{n} from the previous value, we can use the formula to find a more useful way of expressing *M*_{n}. Following the hint of not over-simplifying, use the useful trick of writing *M*_{n} as a series, and look at some values of *M*_{n} where *n* is small: Each *M*_{n} is the sum of a finite geometric series. The exponents in the finite series go from 0 to (*n* – 1). The value of *M*_{n} is the sum of a finite geometric series with *a* = 100, *r* = 0.25, and *n* terms (since the exponents go from 0 to (*n* – 1)). Using the formula for the sum of a finite geometric series, Try these questions, which also ask about the same situation as the example we just saw. |