Problem 77 - Precalculus Mathematics for Calculus

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

Step-by-Step Solution

Answer

(a) Recursive formula: \(S_{1} = 30000\); \(S_{n} = S_{n-1} + 2000\), \(n > 1\). (b) $38,000 in the fifth year.

Step 1: Understanding the Salary Pattern

The salesman's salary increases every year, starting at $30,000 in the first year with an annual raise of $2,000. This means each subsequent year, his salary increases by $2,000 more than the previous year's salary.

Step 2: Formulating Recursive Definition for Salary

A recursive formula defines each term based on the previous term(s). Here, the salary for year \(n\), denoted \(S_{n}\), is based on the salary from the previous year \(S_{n-1}\). Thus, the recursive formula is given by: \[ S_{1} = 30000 \] \[ S_{n} = S_{n-1} + 2000 \text{ for } n > 1 \]

Step 3: Calculating Salary for the Fifth Year

Using the recursive formula to find the salary for the fifth year, calculate each year's salary:- \(S_{1} = 30,000\)- \(S_{2} = S_{1} + 2000 = 30,000 + 2000 = 32,000\)- \(S_{3} = S_{2} + 2000 = 32,000 + 2000 = 34,000\)- \(S_{4} = S_{3} + 2000 = 34,000 + 2000 = 36,000\)- \(S_{5} = S_{4} + 2000 = 36,000 + 2000 = 38,000\)

Step 4: Conclusion

The recursive formula helps us find any year's salary in sequence. Therefore, in the fifth year, the salesman's annual salary will be $38,000.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a type of sequence in which each term after the first is derived by adding a constant difference to the previous term. This consistent difference between consecutive terms is known as the common difference. In the case of the salesman's salary, the series of salaries forms an arithmetic sequence. Each year, his salary increases by a fixed amount of $2,000.
Consider the first few terms of the sales sequence: 30,000; 32,000; 34,000; continued. These numbers form an arithmetic sequence where the common difference, denoted as "d," is 2,000.
This kind of sequence is very systematic, meaning that if you know any term and the common difference, you can find any other term. The formula for the nth term of an arithmetic sequence is given by: \[ a_n = a_1 + (n-1)d \] where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

Salary Progression

Salary progression refers to how an individual's salary increases over time according to a specific pattern or formula. It is essential for understanding long-term income growth, especially when planning personal finances or predicting future earnings.
In our initial problem, the salary progression is linear due to the regular $2,000 incremental annual increase. Starting from a base salary of $30,000, this progression indicates a steady financial growth for the salesman.
This concept is crucial as it allows us to determine future earnings without recalculating each year's salary individually from scratch. It shows a predictable path of income enhancement and helps in making sound financial plans.

Recursive Formula

A recursive formula is a framework where each term is defined as a function of its preceding term(s). It is particularly useful when dealing with sequences where each element can naturally be derived from the previous one.
In the case of the salesman's salary, the recursive formula is structured as follows:\[ S_1 = 30,000 \]\[ S_n = S_{n-1} + 2,000 \] for \( n > 1 \)
Here, \(S_n\) represents the salary in the nth year, while \(S_{n-1}\) symbolizes the salary in the prior year. By knowing the salary of any one year, you can determine the next year's salary simply by adding $2,000. This recursive approach is a methodical way to track salary increments over time, reflecting the clear pattern of annual increases.