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: Identify the problem as a geometric - series problem

Jack's earnings form a geometric series. The first - term $a = 5$, and the common ratio of the geometric series is $r=x$ (expressed as a decimal). The sum of an infinite geometric series is given by $S=\frac{a}{1 - r}$ when $|r|\lt1$, and the series diverges (sum is infinite) when $|r|\geq1$.

Step2: Convert percentages to decimals

$x = 5\%=0.05$, $x = 25\% = 0.25$, $x = 99\%=0.99$, $x = 101\% = 1.01$.

Step3: Determine when the sum of the geometric series is infinite

For a geometric series to have an infinite sum, the common ratio $r$ (where $r$ is the factor by which the payment increases for each additional book) must satisfy $r\geq1$. Among the given values, when $x = 101\%$ (or $r = 1.01$), the sum of the geometric series of Jack's earnings will be infinite as long as he reads more and more books.

Answer:

$x = 101\%$