Question

DISCUSS. DISCOVER: Growth of an Exponential Function

Suppose you are offered a job that lasts one month, and you are to be very well paid. Which of the following methods of payment is more profitable for you?

(a) One million dollars at the end of the month.

(b) Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general,$2^n$ cents on the nth day.

Short answer

The method (b) is more profitable for me.

Step-by-step solution

Step 1. Given.

Two methods of payment:

(a) One million dollars at the end of the month.

(b) Two cents on the first day of the month, 4 cents on the second day, 8 cents on the third day, and, in general,$2^n$cents on the nth day.

Step 2. To determine.

We have to find which method is more profitable for me.

Step 3. Calculation.

In method (a), I’ll get one million dollars $=\$1,000,000$at the end of the month.

In method (b),

Let there are 30 days in a month.

I get,

2 cents on the first day of the month.

$4=2^2$cents on the second day.

$8=2^3$cents on the third day,

and, so on.

$2^{30}$ cents on the 30^th day.

So, my total income is $=2^1+2^2+2^3+\mathrm{.....}+2^{30}$.

Using the sum formula of a geometric series with first term$=2^1=2$

And, $\mathrm{common}\mathrm{ratio}=2\mathrm{and}n=30$terms is:

$$\begin{gathered} 2^1+2^2+2^3+\mathrm{.....}+2^{30} \\ =\frac{2\left(2^{30}-1\right)}{2-1} \\ =\frac{2\left(2^{30}-1\right)}{1} \\ =2\left(2^{30}-1\right) \\ =2.147483646\times {10}^9\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}cents} \end{gathered}$$

Hence, my total income at the end of the month in dollar is:

$$\begin{gathered} =\frac{2.147483646\times {10}^9}{100}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}dollars} \\ =2.147483646\times {10}^7\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}dollars} \\ =21474836.46\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}dollars} \end{gathered}$$

We know that: $\$1,000,000<\$21,474,836.46$.

It means method (b) will be more profitable for me.