Problem 83 - Precalculus: Mathematics for Calculus

Question

A newly hired salesman is promised a beginning salary of \(\$ 30,000\) a year with a \(\$ 2000\) raise every year. Let \(S_{n}\) be his salary in his \(n\) th year of employment. (a) Find a recursive definition of \(S_{n}\) (b) Find his salary in his fifth year of employment.

Step-by-Step Solution

Answer

(a) Recursive formula: \( S_n = S_{n-1} + 2000 \) (b) Salary in fifth year: \( 38000 \)

Step 1: Identify Initial Condition

The initial salary of the salesman is given as $30,000. Hence, \( S_1 = 30000 \). This will serve as the base case in our recursive definition.

Step 2: Understand the Yearly Increment

Every year, the salesman receives a raise of $2,000. This means that each subsequent salary is the previous year's salary plus $2,000.

Step 3: Define the Recursive Formula

Using the information about the increment, we can define the salary for the \( n \)th year as \( S_n = S_{n-1} + 2000 \). This recursive formula accounts for the $2,000 raise every year.

Step 4: Calculate the Fifth Year Salary

We want to find the salary in the fifth year, \( S_5 \). Starting from \( S_1 \) and using our recursive formula: 1. \( S_1 = 30000 \) 2. \( S_2 = S_1 + 2000 = 32000 \) 3. \( S_3 = S_2 + 2000 = 34000 \) 4. \( S_4 = S_3 + 2000 = 36000 \) 5. \( S_5 = S_4 + 2000 = 38000 \). Thus, the salary in the fifth year is \( 38000 \).

Key Concepts

Initial Condition

The concept of an initial condition is like setting the starting line in a race. It tells us where everything begins. In the context of this salesman's salary problem, the initial condition is the salary he starts with. Initially, he was promised a salary of $30,000. This value is crucial because it serves as the base or starting point for all further calculations regarding his salary.

• The initial condition is represented as \( S_1 = 30000 \), denoting his salary in the first year.
• Without the initial condition, you would have no reference point to start calculating future values. Think of it as the zero-yard line in a football game—without it, you wouldn't know how far you've progressed.

Recognizing the initial condition helps in constructing more complex mathematical sequences and recursive patterns by offering a clear starting reference.

Yearly Increment

The yearly increment represents the consistent increment or growth added to a previous value. In salary terms, this is the annual raise promised to the salesman. Each year, he receives an additional $2,000, which means his salary increases by this fixed amount over time.

• This change is crucial for calculating future salaries, as it’s the same amount added each year: \( S_n = S_{n-1} + 2000 \).
• Understanding the increment is necessary for recognizing patterns in sequences. It’s like adding blocks to a tower, where each year (or block) raises the overall height (salary) by a specific amount.

Conceptually, the yearly increment is a key element in the formula that allows us to predict future salaries based on past data, just as understanding annual growth can help a company plan its finances.

Recursive Formula

A recursive formula is a rule that defines each term in a sequence using the previous term(s). For the salary of the salesman, this formula helps calculate his salary for any given year based on the previous one.

• The recursive formula is given as \( S_n = S_{n-1} + 2000 \), where \( S_n \) is the salary for the \( n \)th year.
• This formula starts from the initial condition \( S_1 \) and uses the yearly increment to determine future values.

The beauty of a recursive formula is its ability to build information step-by-step. It's like assembling a chain where each link depends on the one before it. With recursion, you can easily compute the salary for complex sequences of years beyond basic manual calculations. This method is powerful in computer science and mathematics for handling iterative processes efficiently.