College Algebra, Chapter 9, 9.3, Section 9.3, Problem 74

A very patient woman wishes to become a billionaire. She decides to foolow a simple scheme: She puts aside 1 cent the first day, 2 cents the second day, 4 cents the third day, and so on, doubling the number of cents each day. How much money will she have at the end of 30 days? How many days will it take this woman to realize her wish?

If the common ratio is $r = 2$, then the total money that the woman will accumulate after $n = 30$ days is


$
\begin{equation}
\begin{aligned}

S_n =& a \frac{1 - r^n}{1 - r}
\\
\\
S_{30} =& \frac{1 - (2)^{30}}{1 - (2)}
\\
\\
S_{30} =& 1073741823 \text{ cents } \times \frac{1 \text{ dollar}}{100 \text{ cents}}
\\
\\
\text{ or } &
\\
\\
S_{30} =& \$ 10,737,418.23

\end{aligned}
\end{equation}
$


Next, if the woman wishes to have $\$ 1,000,000,000$, then the number of days it will take is..


$
\begin{equation}
\begin{aligned}

1,000,000,000 \text{ dollars } \times \frac{100 \text{ cents}}{1 \text{ dollar}} =& \frac{1 - 2^n}{1-2}
\\
\\
-100,000,000,000 =& 1-2^n
\\
\\
2^n =& 100,000,000,001
\\
\\
n(\ln 2) =& \ln (100,000,000,001)
\\
\\
n =& 36.54 \text{ days or } 37 \text{ days}


\end{aligned}
\end{equation}
$