Problem 107 - Algebra and Trigonometry

Question

You have just signed a 7-year professional football league contract with a beginning salary of \(\$ 2,000,000\) per year. Management gives you the following options with regard to your salary over the 7 years. 1\. A bonus of \(\$ 100,000\) each year 2\. An annual increase of \(4.5 \%\) per year beginning after 1 year 3\. An annual increase of \(\$ 95,000\) per year beginning after 1 year Which option provides the most money over the 7-year period? Which the least? Which would you choose? Why?

Step-by-Step Solution

Answer

Most: Option 3 (\)16,215,000), Least: Option 1 ($14,700,000). Choose Option 3 due to highest total salary.

Step 1: Calculate Total Salary with a Bonus

To find the total salary including a bonus of \(100,000 each year, simply add this bonus to the base salary of \)2,000,000 for each of the 7 years. The calculation is: Total Salary = 7 \times (2,000,000 + 100,000) = 7 \times 2,100,000 = 14,700,000.

Step 2: Calculate Total Salary with an Annual Increase of 4.5%

To find the total salary including an annual increase of 4.5% per year, start with the base salary and apply the increase cumulatively year after year. Year 1 Salary: \(2,000,000 Year 2 Salary: \)2,000,000 \times 1.045 Year 3 Salary: (2,000,000 \times 1.045^2) And so on...Use the formula for sum of geometric series to find the total over 7 years: \(\text{Total Salary} = 2,000,000 \times \frac{1-1.045^7}{1-1.045} = 15,672,399.58 \)

Step 3: Calculate Total Salary with an Annual Increase of \(95,000

Calculate the total salary including an annual increase of \)95,000 per year: Year 1 Salary: \(2,000,000 Year 2 Salary: \)2,000,000 + 95,000 Year 3 Salary: \(2,000,000 + 95,000 \times 2 And so on...Use the arithmetic series sum formula to find the total over 7 years: \(a = 2,000,000 \)\(d = 95,000 \)\(n = 7 \)\(\text{Total Salary} = \frac{n}{2} \times (2a + (n-1) \times d) = 2,000,000 \times 7 + \frac{7}{2} \times 6 \times 95,000 = 16,215,000 \)

Key Concepts

arithmetic series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference, denoted by \(d\). For example, if you start with 2 and keep adding 3, you get 2, 5, 8, 11, and so on.
To find the sum of an arithmetic series, you can use the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1) \times d) \]
Where:

• \(S_n\) is the sum of the first \(n\) terms
• \(a\) is the first term
• \(d\) is the common difference
• \(n\) is the number of terms

In the football contract example, the third option involves an arithmetic series. The initial salary is \(2,000,000\), the common difference is \(95,000\), and the number of terms is 7. By plugging these values into the formula, we get the sum over the 7 years.

geometric series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, if you start with 1 and multiply by 2, you get 1, 2, 4, 8, and so on.
To find the sum of the first \(n\) terms of a geometric series, you can use the formula:
\[ S_n = a \times \frac{1-r^n}{1-r} \] Where:

• \(S_n\) is the sum of the first \(n\) terms
• \(a\) is the first term
• \(r\) is the common ratio
• \(n\) is the number of terms

In the football contract example, the second option involves a geometric series. The initial salary is \(2,000,000\), and the common ratio with an annual increase of 4.5% is \(1.045\). By plugging these values into the formula, we get the total salary over 7 years.

financial mathematics

Financial mathematics is the application of mathematical methods to financial problems. It involves understanding interest rates, investments, annuities, and more. In salary calculations, financial mathematics helps us determine the best financial choices by comparing different salary structures.
In our football contract example, financial mathematics allows us to use arithmetic and geometric series formulas to calculate total earnings for different payment options. This helps the player choose the most beneficial option.

annual salary increase

An annual salary increase is a rise in salary that occurs every year. It can be a fixed amount or a percentage of the current salary. This affects overall earnings significantly over time.
In the football contract example, the first option offers a fixed annual bonus of \(100,000\). The second option offers a percentage increase of 4.5% per year. The third option offers an annual increase of \(95,000\).

Choosing the best salary option depends on calculating the total salary over 7 years for each option and comparing them. Financial mathematics helps us understand which option provides the most money and why.