Problem 59 - Algebra and Trigonometry

Question

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 \(n\) th day

Step-by-Step Solution

Answer

Option (b) is more profitable with approximately $21,474,836.46 earned.

Step 1: Understand the Payment Options

First, let's understand the two payment options: 1. A lump sum of one million dollars at the end of the month. 2. An exponential payment where you receive 2^n cents on the nth day, for each day of the month.

Step 2: Convert Cent to Dollars

Since the payment is given in cents, we need to convert it to dollars for easy comparison. 1 cent = 0.01 dollars. The payment on the nth day in dollars is: \[ \text{Payment in Dollars} = \frac{2^n}{100} \text{ dollars} \]

Step 3: Calculate Total Exponential Earnings

To calculate the total earnings for the exponential payment scheme, we calculate the sum for each day from 1 to 30 (assuming a 30-day month) using the formula for the sum of a geometric series: \[ S_n = a \frac{r^n - 1}{r - 1} \] where \(a = 0.02\) dollars, \(r = 2\), and \(n = 30\). Therefore, the total earnings are: \[ S_{30} = 0.02 \frac{2^{30} - 1}{1} \]

Step 4: Calculate and Compare

Calculate the total earnings for the exponential scheme using the formula: \[ S_{30} = 0.02 (2^{30} - 1) \] Simplify the calculation:\[ S_{30} = 0.02 \times (1073741824 - 1) = 0.02 \times 1073741823 \] \[ S_{30} = 21474836.46 \] Since the total earnings from the exponential method over 30 days is approximately \(21,474,836.46, compare it with the one million dollar lump sum, i.e., \)1,000,000.

Step 5: Draw Conclusion

By comparing the two payment methods, the exponential payment grows rapidly and exceeds the lump sum by the end of the month with earnings of approximately $21,474,836.46 versus $1,000,000 from option one.

Key Concepts

Understanding Geometric Series

A geometric series is a mathematical concept where each term in the series is a fixed multiple of the term before it. This is an important foundation for understanding many phenomena that grow at a consistent rate.

In the context of the payment scenario, each day's payment doubles, making it a perfect example of a geometric series. To break it down, let's consider the first few terms:

• On the first day, you receive 2 cents.
• On the second day, it doubles to 4 cents.
• On the third day, it doubles again to 8 cents.

This doubling process continues every day, following the function expressed by the formula: \[ a r^{n-1} \] where \( a \) is the initial term (0.02 dollars or 2 cents), \( r \) is the common ratio (2 in this situation), and \( n \) represents the day number. By knowing how geometric series operate, you can predict substantial growth over several terms, which helps in deciding between different financial strategies.

Exponential Function in Depth

An exponential function is a mathematical expression where the variable is in the exponent. This kind of function is crucial in modeling situations where growth accelerates rapidly over time, such as population growth, radioactive decay, or, in this case, financial growth.

For the job payment scheme, the formula \[ 2^n \] characterizes daily payments. Here, 'n' refers to each day of the month, showing how exponential growth quickly leads to large sums.The magic of exponential functions lies in their compounding nature. At first, growth seems slow, but it speeds up over time. Notably, this rapid increase becomes apparent when you calculate the total amounts for each subsequent day. By day 30, this exponential growth results in payments that far exceed the initial modest sums. This characteristic makes exponential functions both powerful and sometimes unintuitive in terms of their results over time.

Financial Comparison of Payment Methods

In financial decision-making, comparing different payment structures is key to maximizing benefits. In our exercise, we have two payment methods: a lump sum of one million dollars at the end of the month or exponential growth of daily payments.

This scenario highlights how small, regularly compounded increments can surpass a seemingly large fixed sum through exponential growth.

• The lump sum option is simple: you know exactly what you are going to get.
• In contrast, the exponential payment option, while initially less attractive, ends up totaling over 21 million dollars by month’s end.

The choice between these two involves understanding and applying the concept of time value of money. Exponential growth often holds the upper hand in long-term scenarios, as the payment compounding impact leads to significant returns compared to fixed sums. With foresight and careful comparison, you can see how exponential methods not only compete with but vastly outstrip large initial amounts, illustrating the vital nature of understanding financial growth patterns.