Problem 77 - Algebra and Trigonometry

Question

Salary Increases 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 \(S_{m}\) (b) Find his salary in his fifth year of employment.

Step-by-Step Solution

Answer

(a) Recursive: \(S_{n} = S_{n-1} + 2,000\), \(S_1 = 30,000\); (b) Fifth year salary: \(38,000\) dollars.

Step 1: Defining the Initial Salary

The initial condition for this problem is that the salesman starts with a salary of \(S_1 = 30,000\) dollars in his first year. This establishes the base case for the recursive definition of his salary.

Step 2: Understanding the Annual Raise

Every year, the salesman receives a raise of \(2,000\) dollars. This consistent annual increment will be key to developing the recursive formula that describes the salary for subsequent years.

Step 3: Formulating the Recursive Definition

A recursive definition for the salary can be written as follows: \( S_{n} = S_{n-1} + 2,000 \) for \(n > 1\), with an initial condition \(S_1 = 30,000\). This equation summarizes how the salary increases each year based on the salary from the previous year.

Step 4: Calculating the Salary for Fifth Year

To find the salary in the fifth year, we use the recursive relation step by step: \(S_1 = 30,000\), \(S_2 = S_1 + 2,000 = 32,000\), \(S_3 = S_2 + 2,000 = 34,000\), \(S_4 = S_3 + 2,000 = 36,000\), \(S_5 = S_4 + 2,000 = 38,000\). Thus, the salary in the fifth year, \(S_5\), is \(38,000\) dollars.

Key Concepts

Initial Condition

When solving recursive sequences, the initial condition is crucial as it sets the starting point for all subsequent calculations. In this example, the initial condition refers to the salesman's starting salary, which is given as $30,000 in his first year of employment. This initial salary is denoted as \( S_1 = 30,000 \).
Without this base value, we wouldn't have a specific point from which to apply the formulas or recursively calculate future salaries. The initial condition acts as a foundation or benchmark that ensures consistent calculations throughout the sequence.
Therefore, when tasked with similar exercises involving recursive sequences, identifying and clearly stating the initial condition will always be your first step. It grounds the whole problem and makes sure your recursive formulas work as intended.

Annual Increment

An annual increment is the increase added to each subsequent term or instance in a recursive formula. In real-life scenarios, it might be analogous to a steady raise or growth step. Here, the salesman receives a consistent raise of $2,000 every year. This means his salary increases by this amount from one year to the next.
Understanding the concept of annual increment is essential because it represents the change factor in a recursive series. It is the constant that motivates the yearly salary progression in this context.
Recognizing this increment enables you to apply this knowledge to each year, using it to bridge one salary amount to the next. This ultimately helps in forming a solid recursive formula, which can universally apply to any year's salary calculation in this sequence.

Recursive Formula

A recursive formula defines a sequence in terms of previous terms. It's like a pattern that shows you how to get from one step to the next using a rule. In this salary example, the recursive formula is written as \( S_{n} = S_{n-1} + 2,000 \) for \( n > 1 \).
This means the salary for any year \( n \) can be calculated by taking the salary from the previous year \( (S_{n-1}) \) and adding the annual increment of $2,000. The simplicity of the recursive formula is one of its strengths, as it provides an efficient way to compute terms in a sequence without directly calculating each term from the start.
Understanding recursive formulas is vital for tackling numerous mathematical and real-world problems and efficiently calculating future predictions or events. Always ensure you comprehend the link between each term and its predecessor, as this connection is the essence of recursion in sequences.