Problem 89 - Precalculus with Limits

Question

In Exercises 89 and 90, consider a job offer with the given starting salary and the given annual raise. (a) Determine the salary during the sixth year of employment. (b) Determine the total compensation from the company through six full years of employment. Starting Salary \( \$32,500 Annual Raise \) \$ 1500 $

Step-by-Step Solution

Answer

The salary during the sixth year of employment is \$40,000 and the total compensation from the company through six full years of employment is \$217,500.

Step 1: Calculate the Salary for the 6th Year

The starting salary is \$32,500. Each year, there is an annual raise of \$1500. To find the salary during the sixth year, find the starting salary and add the raise increased annually for 5 years. The formula to find the sixth year's salary is: Starting salary + (Annual raise * Number of years). The number of years is 5 because the starting salary is for the first year: \( \$32,500 + (\$1500 * 5) = \$32,500 + \$7500 = \$40,000 \). So, the salary during the sixth year of employment is \$40,000.

Step 2: Calculate the Total Compensation for 6 years

To calculate total salary paid over six years, use the formula for the sum of an arithmetic series since the annual salary forms an arithmetic sequence. The formula is \( S = n/2 * (a_1 + a_n) \) where \( n \) is the number of terms, \( a_1 \) is the first term (starting salary), and \( a_n \) is the last term (salary of 6th year). So, the total salary earned over 6 years is: \( 6/2 * (\$32,500 + \$40,000) = 3 * \$72,500 = \$217,500 \).

Step 3: Review the Results

Look over both answers to ensure they make sense. The salary for the sixth year of employment is \$40,000 and the total compensation over six years is \$217,500.

Key Concepts

Annual Salary Increase

An annual salary increase is a raise in salary that occurs each year, often as a recognition of the employee's service and to match inflation rates. In our exercise, this increase is \(1,500 each year. Such consistent increments signify a pattern known as an arithmetic sequence in mathematics.
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms, known as the common difference, remains constant. So, if you start with a certain salary,

• the first year you get your base salary,
• the second year, your salary is the base salary plus \)1,500,
• the third year, it becomes base salary plus \(3,000,
• and so on.

By the time you're in your sixth year, the accumulated raise is \(5 \times 1500 = 7500\), which when added to the initial salary, gives you the sixth year's salary of \)40,000.

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. If an employee receives a consistent raise each year, the sequence of annual salaries can be represented as an arithmetic sequence. In our problem, the task was to find the sum of all salaries over 6 years, making it an arithmetic series problem.
The formula to find the sum of an arithmetic series is \[ S = \frac{n}{2} \times (a_1 + a_n) \]where:

• \( n \) is the number of terms, which is 6 in this exercise,
• \( a_1 \) is the first term or the salary at the start of the employment, \(32,500,
• \( a_n \) is the last term, the salary in the sixth year, \)40,000.

Thus, applying the formula gives us the total compensation over six years as \( 3 \times (32,500 + 40,000) = 3 \times 72,500 = 217,500\).
This sum represents the total amount earned by the employee with regular annual increases.

Total Compensation

Total compensation refers to the complete salary an employee receives over a specified period, including all raises and bonuses. In the exercise, this involved calculating the total salary over six years of employment. Estimating total compensation is crucial for understanding job offers and financial planning.
For the given scenario, starting with a salary of $32,500 and incrementing annually by $1,500, the total compensation was calculated using the arithmetic series formula, leading to a total of $217,500 at the end of six years.

• This sum accounts for the first year's salary plus all subsequent increments over the tenure.
• Understanding total compensation helps in evaluating job offers and long-term financial commitments.

By successfully finding this total, employees can see the full financial benefit of consistent raises throughout their tenure.