Problem 75 - Algebra and Trigonometry

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) \( S_1 = 30000, \; S_{n} = S_{n-1} + 2000 \); (b) \( S_5 = 38000 \)

Step 1: Understand the Problem

We need to define a recursive formula for the salary, starting with an initial amount and increasing it by a fixed amount each year. Additionally, we will calculate the salary for the fifth year.

Step 2: Define the Initial Condition

The initial salary, or the salary in the first year, is given by \( S_1 = 30000 \). This sets the base for our recursion.

Step 3: Define the Recursive Formula

Since the salary increases by \(2000\) each year, the recursion can be represented as \( S_{n} = S_{n-1} + 2000 \) for \( n \geq 2 \).

Step 4: Calculate the Salary for the Fifth Year

Using the recursive formula, we calculate:1. \( S_2 = S_1 + 2000 = 30000 + 2000 = 32000 \)2. \( S_3 = S_2 + 2000 = 32000 + 2000 = 34000 \)3. \( S_4 = S_3 + 2000 = 34000 + 2000 = 36000 \)4. \( S_5 = S_4 + 2000 = 36000 + 2000 = 38000 \)Therefore, the salary in the fifth year is \(38000\).

Key Concepts

Initial Condition

The "initial condition" is an essential component when dealing with sequences or recursive formulas. It sets the starting point for the sequence. In the context of our salary problem, the initial condition refers to the salary value at the beginning of the first year.
For the newly hired salesman, it's given as \( S_1 = 30,000 \). This means that in the very first year, the salesperson earns \(30,000\) dollars.
Why is the initial condition important? It ensures that there's a known starting value for calculating subsequent values in the sequence. Without an initial condition, we would not have a base reference to apply further calculations.

• Think of it as the anchor point that the entire salary progression rests upon.
• In recursive problems, the initial condition gives us the specific number to begin applying the recursion.

Whether you are defining a sequence for salary calculation or a progression in a sequence of numbers, always start by identifying the initial condition.

Recursive Formula

A "recursive formula" is a way to define the successive terms of a sequence using some of its previous terms. It breaks down the process of finding a term into smaller and more manageable steps.
In our salary calculation example, the recursive formula is used to determine the salesman’s salary each year:
\( S_n = S_{n-1} + 2000 \) for \( n \geq 2 \).
This equation tells us everything we need: to find the salary in the \( n \)th year, simply take the salary of the previous year (\( S_{n-1} \)) and add \(2,000\) dollars to it.
Why do we use recursive formulas?

• They simplify complex calculations. By using the result from the previous step, these methods allow step-by-step computation.
• They can highlight patterns within the sequence.
• Given the initial condition, anyone can calculate further terms without needing additional information.

Recursive formulas are widely used in various domains, including mathematics and computing, due to their efficiency and simplicity.

Salary Calculation

Using the concepts of the initial condition and recursive formula, we can calculate the salary over multiple years. In this scenario, the question asks for the salary in the fifth year of our salesman’s employment.
Starting with the initial condition, where the salary in the first year is \(30,000\) dollars, we can simply apply the recursive formula to find future salaries. With each year, the salary increases steadily by \(2,000\) dollars.
Here's how:

• Year 2: \( S_2 = 30,000 + 2,000 = 32,000 \)
• Year 3: \( S_3 = 32,000 + 2,000 = 34,000 \)
• Year 4: \( S_4 = 34,000 + 2,000 = 36,000 \)
• Year 5: \( S_5 = 36,000 + 2,000 = 38,000 \)

In the fifth year, the salesman’s salary amounts to \(38,000\) dollars. By understanding the recursive application and knowing where to start with the initial condition, you can determine any year's salary with ease.