Problem 75 - Precalculus: Mathematical 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 definition: \( S_{n} = S_{n-1} + 2000 \) for \( n > 1 \). (b) In the fifth year, the salary is \( 38000 \).

Step 1: Understanding the Problem

The question involves a salesman whose starting salary is $30,000, with an annual raise of $2,000. We need to express this as a sequence where each term represents the salary after each year, and create a recursive formula for it.

Step 2: Defining Initial Conditions

The initial condition is the salary in the first year, which is given as \( S_1 = 30000 \). This forms the base case for our recursive sequence.

Step 3: Establishing the Recursive Formula

The salary each year is the previous year's salary plus the $2000 raise. Thus, the recursive formula is \( S_{n} = S_{n-1} + 2000 \) for \( n > 1 \).

Step 4: Calculating Salary for the Fifth Year

We calculate each year's salary using the recursive formula until the fifth year:- \( S_1 = 30000 \)- \( S_2 = S_1 + 2000 = 32000 \)- \( S_3 = S_2 + 2000 = 34000 \)- \( S_4 = S_3 + 2000 = 36000 \)- \( S_5 = S_4 + 2000 = 38000 \).Thus, in the fifth year, the salary \( S_5 = 38000 \).

Key Concepts

Initial Conditions

When working with recursive sequences, the initial condition is a crucial piece of information. It sets the starting point or base case for the sequence.

In this scenario, the initial condition is the salesman's starting salary, which is given as $30,000. This is represented in the sequence as:

• For the first year, the salary is denoted as \( S_1 = 30000 \).

This initial condition is essential because it forms the basis from which all subsequent terms in the sequence are derived.

Put simply, without the initial condition, we wouldn't know where to begin calculating the salaries for the following years.

Recursive Formula

The recursive formula is the rule or process that generates each term of the sequence from the previous term. It is a powerful method, especially when dealing with problems where each term depends on the ones that come before it.

In the context of the salesman's salary, the recursive formula provides a simple way to determine his salary for each successive year.

• The formula is expressed as \( S_{n} = S_{n-1} + 2000 \) for \( n > 1 \).

This implies:

• Each year's salary is the previous year's salary plus a $2,000 raise.

This recursive formula makes it straightforward to calculate the salary for any year, once the initial condition is known.

By following this pattern, we can predict future salaries based on past data without performing complex calculations.

Salary Calculation

The process of computing the salary in a specific year is an application of the recursive sequence. Utilizing the recursive formula, we can "build up" to the salary of interest.

Let's see how it works for the salesman's salary by detailing the steps to reach his salary in the fifth year:

• Start with the initial condition: \( S_1 = 30000 \).
• Apply the recursive formula to find each subsequent year's salary:
  • Second year: \( S_2 = S_1 + 2000 = 32000 \).
  • Third year: \( S_3 = S_2 + 2000 = 34000 \).
  • Fourth year: \( S_4 = S_3 + 2000 = 36000 \).
  • Fifth year: \( S_5 = S_4 + 2000 = 38000 \).

By iterating through these calculations, we find that the salary in the fifth year is \( S_5 = 38000 \). This step-by-step approach illustrates the power and simplicity of using recursive formulas for salary calculation over time, especially when an annual increment is involved.