Exercise 1.4.22

Question

Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer?\begin{enumerate}[label=\Roman*.]
\item One million dollars at the end of the month.
\item One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general, $2^{n-1}$ cents on the nth day.
\end{enumerate}

Khagan Gulusoy · Accepted Answer

Note that the sequence in II is a geometric progression. Therefore, if we note $S_{n}$ as the sum of $n$ terms of the given geometric progression, we have:

$$
\forall n \in[0,31]: S(n)=\frac{a\left(1-r^{n}ight)}{1-r} 	ext {, }
$$

where $r$ is the common ratio, and $a$ is the first term of the progression. For $n=30$ (one month), we have:

$$
S_{30}=\frac{1\left(1-2^{30}ight)}{1-2}=2^{30}-1=1013741823,
$$

for $n=31$, we have:

$$
S_{31}=\frac{1\left(1-2^{31}ight)}{1-2}=2^{31}-1=212440364
$$

for $n=28$, we have:

$$
S_{28}=\frac{1\left(1-2^{28}ight)}{1-2}=2^{28}-1=20833455 	ext {, }
$$

for $n=29$, we have:

$$
S_{29}=\frac{1\left(1-2^{29}ight)}{1-2}=2^{29}-1=558609011
$$

Since $S_{30}>10^{6}, S_{31}>10^{6}, S_{28}>10^{6}, S_{25}>10^{6}$, it would be more profitable to choose the second option.

Exercise 1.4.22

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Exercise 1.4.22

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