Chapter 12.3, Problem 63SR

To determine

To find: How much would the second option be worth.

Answer to Problem 63SR

The worth of second option is $\$10737418.23$ .

Explanation of Solution

Given information:

The king gave the boy a choice.

He could have $\$1,000,000$ at once, or he could be rewarded daily for a $30$ -day month,

With One penny on the first day, two pennies on the second day, and so on, receiving twice as many pennies each day as the previous day.

Given

With One penny on the first day, two pennies on the second day, and so on, receiving twice as many pennies each day as the previous day.

It means it follows a geometric progression.

  $1,2,2^2,\mathrm{.....}$

We know

Sum of $n$ terms of geometric progression $S_n=\frac{a\left(r^n-1\right)}{r-1}$

Here

  $\begin{array}{l}n=30\\ a=1\\ r=2\end{array}$

On substituting values

We get

  $\begin{array}{l}{S}_n=\frac{a\left(r^n-1\right)}{r-1}\\ \Rightarrow S_n=\frac{1\left(2^{30}-1\right)}{2-1}\\ \Rightarrow S_n=2^{30}-1\\ \Rightarrow S_n=1073741824-1\\ \Rightarrow S_n=1073741823\end{array}$

Therefore,

The worth of second option is $1073741823$ pennies.

We know

  $1\$=100$ pennies

Using that

We get

The worth of second option is $\$10737418.23$ .