Chapter 12.3, Problem 78E

To determine

The amount of money at the end of 30th days and number of days it takes women to realize her wish.

Answer to Problem 78E

The total amount she saves at the end of 30th day is $536870911$ and the number of days to become a billionaire is $31$.

Explanation of Solution

Given:

The women wishes to become a billionaire and she saves the cents in a geometric sequence like 1,2,4…

Calculation:

The value of $r$ in the given situation is $2$.

Value of $n$ is $30$.

Find the value of $S_{30}$,

$\begin{array}{c}{S}_{30}=\frac{a\left(1-r^{29}\right)}{1-r}\\ =\frac{1\left(1-{\left(2\right)}^{29}\right)}{1-2}\\ =536870911\end{array}$

Assume she saved 1073741823 cents to become a billionaire.

Find number of days it takes the women to realize her wish,

$S_n=\frac{a\left(1-r^{n-1}\right)}{1-r}$

Substitute 1073741823 for $S_n$,

$\begin{array}{c}\text{1}0\text{73741823}=\frac{1\left(1-2^{n-1}\right)}{1-2}\\ \text{1}0\text{73741824}={\text{2}}^{n-1}\\ 2^{30}={\text{2}}^{n-1}\\ n=31\end{array}$

Thus the total amount she saves at the end of 30th day is $536870911$ and the number of days to become a billionaire is $31$.