那位数学大侠给解答孩子的问题， 谢！
Recurrence relations：
You’ve won a contest! You’re going to win money! Your prize is determined as follows. You are given $40, then asked to sit in a chair. At each minute mark of you being in the chair, your winnings are re-calculated as being 150% of the amount you held during the previous minute but deducted from that is 25% of the amount you held the minute before that (note that you held $0 before the contest started). Whoever is holding the contest is no fool; it’s not hard to see that there needs to be some cost to you sitting in the chair, or they’ll go bankrupt! So, at each minute mark, you’re going to lose $6 for every minute you’ve been in the chair (after the first minute you’ll lose $6, after the second minute you’ll lose an another $12, after the third minute you’ll lose another $18, and so on). You can leave the chair any time you want, collect your winnings, and walk away.
a) Write a new recurrence relation that expresses the amount of money you win if you leave the chair after the mth minute (but before minute m + 1).
b) Solve this recurrence relation to find an explicit function of m for your winnings after m minutes.
c) How long should you stay in the chair to maximize your winnings? If you make any claims about the behaviour of the function after a given point, make sure you justify your answer (this can be done using basic calculus or by other means).
Recurrence relations：
You’ve won a contest! You’re going to win money! Your prize is determined as follows. You are given $40, then asked to sit in a chair. At each minute mark of you being in the chair, your winnings are re-calculated as being 150% of the amount you held during the previous minute but deducted from that is 25% of the amount you held the minute before that (note that you held $0 before the contest started). Whoever is holding the contest is no fool; it’s not hard to see that there needs to be some cost to you sitting in the chair, or they’ll go bankrupt! So, at each minute mark, you’re going to lose $6 for every minute you’ve been in the chair (after the first minute you’ll lose $6, after the second minute you’ll lose an another $12, after the third minute you’ll lose another $18, and so on). You can leave the chair any time you want, collect your winnings, and walk away.
a) Write a new recurrence relation that expresses the amount of money you win if you leave the chair after the mth minute (but before minute m + 1).
b) Solve this recurrence relation to find an explicit function of m for your winnings after m minutes.
c) How long should you stay in the chair to maximize your winnings? If you make any claims about the behaviour of the function after a given point, make sure you justify your answer (this can be done using basic calculus or by other means).
