Problem 133 - Precalculus with Limits

Question

An investment firm has a job opening with a salary of \( \$45,000 \) for the first year. Suppose that during the next \( 39 \) years, there is a \( 5\% \) raise each year.Find the total compensation over the \( 40 \)-year period

Step-by-Step Solution

Answer

The total compensation over the 40-year period will be approximately \$4,745,104.09.

Step 1: Define the variable and formula

This situation is a perfect example of a geometric series, where each term after the first is found by multiplying the previous term by a fixed non-zero number. In our case this fixed number is \(1.05\) (because of the 5% increase every year). The first term of the sequence is \( \$45,000 \). The formula for the sum \(S\) of a finite geometric series is \(S = a \times \frac{1 - r^n}{1 - r}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.

Step 2: Insert the given values into formula

By inserting \(a = \$45,000\), \(r = 1.05\), and \(n = 40\) into our formula, it becomes: \(S = \$45,000 \times \frac{1 - (1.05)^{40}}{1 - 1.05}\).

Step 3: Calculate the total sum

After performing the calculations in above step, the total compensation received over the 40-year period is found out to be approximately \$4,745,104.09.

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant value known as the common ratio. This concept is widely used in various calculations, such as salary increments or population growth.
A geometric sequence is represented mathematically as:

• The first term: \( a \)
• The common ratio: \( r \)
• The \( n^{th} \) term: \( a \cdot r^{n-1} \)

For instance, in the context of the original exercise concerning salary increases, the first-year salary is the first term \( a = \$45,000 \), and the annual increment of 5% is the common ratio \( r = 1.05 \). Each subsequent year's salary can be determined by multiplying the previous year's salary by 1.05. This structure of repeatedly increasing values fits perfectly into the framework of a geometric sequence.

Compound Interest

The concept of compound interest is closely related to geometric sequences. In finance, compound interest occurs when interest is added to the principal sum so that from that moment, the interest added also earns interest. This results in exponential growth, similar to how salaries increase in a geometric sequence.
Compound interest can be calculated using the formula:

• The principal amount: \( P \)
• The interest rate: \( r \)
• The number of compounding periods: \( n \)
• The formula: \( A = P(1 + r)^n \)

This formula shows that the investment amount \( A \) grows exponentially with each compounding period. In the context of salaries, each yearly increment functions similar to compounding periods, compounding the initial salary based on a fixed percentage.

Series Sum Formula

The series sum formula is a powerful tool to calculate the total value of all terms in a geometric sequence. Understanding how to use it can save you time and ensure accuracy in your calculations.
For a finite geometric series, the sum \( S \) is calculated with the formula:

• First term: \( a \)
• Common ratio: \( r \)
• Number of terms: \( n \)
• Formula: \( S = a \times \frac{1 - r^n}{1 - r} \)

This formula allows you to find the total salary over a period of time, considering annual raises, as demonstrated in the given exercise. By substituting the relevant values into the formula, the total compensation over the forty-year career was calculated, revealing how the series sum formula simplifies the complex process of considering each year's increase individually.