Question

Jill has 64 ounces of lemonade. Each day, her brother drinks 16 ounces. Then at the end of each day Jill adds 25% of the remaining total back into the lemonade container. Which recursive formula models the number of ounces of lemonade in the container, an, at the end of the nth day?
1 , 2 , 3 , 4

Answer 1

Let a_0 = 64 (ounces at the start, before any days). Each day the brother drinks 16, leaving a_{n-1} − 16, then Jill adds 25% of that, so the end amount is 1.25 times the remainder. Thus for n ≥ 1:

a_n = (5/4)(a_{n-1} − 16) = (5/4)a_{n-1} − 20.

(If you index so a_1 is the amount at the end of day 1, then a_1 = 60 and for n ≥ 2 use the same recurrence: a_n = (5/4)(a_{n-1} − 16).)