Question

to encourage jack to read during the summer, his father offers an incentive. he will pay jack $5 for the first book he reads. for each additional book jack reads, his father will pay him an amount that is equal to some percentage, x, of the amount he paid for the previous book. jacks father agrees to let jack choose a value for x, and jack would obviously like his total earnings to be as large as possible. which of these options will allow jack to earn an unlimited amount of money from his father, as long as he reads more and more books? x = 5% x = 25% x = 99% x = 101%

Explanation:

Step1: Analyze the problem concept

We want to maximize the total earnings with an infinite - number of books read. If the payment for the \(n\)th book is \(a_n\), and \(a_1 = 5\), and \(a_{n+1}=x\times a_n\). The sum of an infinite geometric series \(S=\sum_{n = 1}^{\infty}a_n\) (when \(|r|\lt1\), the sum of an infinite geometric series \(\sum_{n=0}^{\infty}ar^n=\frac{a}{1 - r}\), here our series starts from \(n = 1\) and \(a=a_1 = 5\) and \(r=x\)) is \(S=\frac{5}{1 - x}\) for \(x\lt1\). But if \(x\geq1\), the sum of the series will be infinite.

Step2: Evaluate the options

We want to make the sum of the payments for an unlimited number of books as large as possible. If \(x = 101\%=1.01\), as Jack reads more books, the amount he gets for each new book is larger than the previous one, and the total amount he can earn will be infinite. For \(x = 25\% = 0.25\), \(x=5\%\) and \(x = 99\%=0.99\), the sum of the geometric - series of payments will be a finite value \(\frac{5}{1 - x}\).

Answer:

\(x = 101\%\)