Problem 73 - Algebra and Trigonometry

Question

Use the formula for the sum of the first n terms of a geometric sequence to solve Exercises \(71-76\). A job pays a salary of \(\$ 24,000\) the first year. During the next 19 years, the salary increases by \(5 \%\) each year. What is the total lifetime salary over the 20 -year period? Round to the nearest dollar,

Step-by-Step Solution

Answer

The total lifetime salary over the 20-year period is approximately \$ 987,754.

Step 1: Formulate the Geometric Sequence

In this case, we have a geometric sequence where the first term \(a_1\) is the initial salary, \$24,000, and the common ratio \(r\) is 1.05. This is because the salary increases by 5 percent every year.

Step 2: Use the formula for the sum of a geometric sequence

We will use the sum of a geometric sequence formula \(S_n = a_1\frac{1-r^n}{1-r}\) where \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, \(r\) is the common ratio and \(n\) is the number of terms. In our case, \(n = 20 \), as the salary progression is set for 20 years.

Step 3: Calculate the total sum

Plugging values into the formula we get: \( S_{20} = 24000 \times \frac{1-(1.05)^{20}}{1-1.05} \)

Step 4: Compute and Round

Calculate your answer and round it to the nearest dollar. The result should be in a dollar format.

Key Concepts

Geometric Sequence

A geometric sequence is a series 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 instance, in a job salary scenario where an employee's yearly salary starts at $24,000 and increases by 5% annually, the salary for each subsequent year can be represented as a geometric sequence.

Sum of a Geometric Series

When faced with the task of calculating the total amount of money an employee earns over a certain period of time with a constant percentage increase per year, you would use the sum of a geometric series formula. In our example, the formula is \( S_n = a_1\frac{1-r^n}{1-r} \), where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the initial salary, \( r \) is the common ratio representing the annual growth rate of the salary, and \( n \) stands for the number of years. By substituting the known values into this formula, we can calculate the total lifetime salary over the specified period.

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to the current value, resulting in its growth at a consistently increasing rate. This concept is perfectly exemplified in the context of a salary increasing at a constant percentage every year. In such a case, the salary doesn't just increase by a fixed amount each year; rather, it grows by a percentage, meaning each year's increase is larger than the last if the salary amount increases. Over time, this leads to significantly larger figures than a simple linear progression, embodying the power of exponential growth.