Problem 63 - Precalculus Mathematics for Calculus

Question

A man gets a job with a salary of \(\$ 30,000\) a year. He is promised a \(\$ 2300\) raise each subsequent year. Find his total earnings for a 10 -year period.

Step-by-Step Solution

Answer

The total earnings over 10 years is \( \$ 403,500 \).

Step 1: Understand the Sequence of Earnings

The man's earnings form an arithmetic sequence where the first term, \( a_1 \), is \( 30,000 \), and the common difference, \( d \), is \( 2,300 \).

Step 2: Identify the Formula for the Total Earnings

The sum of an arithmetic sequence can be found using the formula: \[ S_n = \frac{n}{2} \times (2a_1 + (n-1) \cdot d) \] where \( n \) is the number of terms, \( a_1 \) is the first term, and \( d \) is the common difference.

Step 3: Substitute the Known Values into the Formula

Substitute \( n = 10 \), \( a_1 = 30,000 \), and \( d = 2,300 \) into the formula: \[ S_{10} = \frac{10}{2} \times (2 \times 30,000 + (10 - 1) \times 2,300) \]

Step 4: Calculate the Inside Components of the Formula

Calculate the components inside the parentheses: \[ 2 \times 30,000 = 60,000 \] \[ 9 \times 2,300 = 20,700 \] Thus, the inside of the parentheses becomes \( 60,000 + 20,700 = 80,700 \).

Step 5: Solve for the Total Earnings

Now calculate \( S_{10} \): \[ S_{10} = \frac{10}{2} \times 80,700 = 5 \times 80,700 = 403,500 \]

Step 6: Conclusion

The man's total earnings over the 10-year period is \( 403,500 \).

Key Concepts

Total Earnings

Total earnings refer to the sum of all income received over a particular period. In this example, we consider a man's salary over ten years. Each year, his salary increases by a specific amount, forming a pattern known as an arithmetic sequence.
To calculate total earnings, you need to know:

• The initial annual salary - here, it's \(30,000\).
• The annual raise - here, it's \(2,300\).
• The number of years he works - here, it's \(10\ years\).

Understanding these three components helps us determine the overall income by accumulating each year's salary. As the sequence of earnings progresses with a consistent increase, the calculation becomes straightforward using arithmetic sequence concepts.

Sum of Sequence

The sum of a sequence is a fundamental aspect when dealing with arithmetic sequences, especially when calculating total earnings over time. Here, the sequence is defined by annual salary increases, starting from a set amount and increasing by a constant value.
To find the sum of this sequence, we utilize a formula specifically designed for arithmetic sequences. The formula is:
\[ S_n = \frac{n}{2} \times (2a_1 + (n-1) \cdot d) \]
where:

• \( S_n \) is the sum of the first \( n \) terms.
• \( n \) represents the number of terms - in this case, \(10\ years\).
• \( a_1 \) is the first term - \(30,000\).
• \( d \) is the common difference between terms - \(2,300\).

By substituting these values into the formula, we calculate the total earnings over the ten-year period. This formula shows us how to efficiently sum an entire decade of income, taking all salary increases into account.

Arithmetic Progression

An arithmetic progression (also known as an arithmetic sequence) is a sequence of numbers where each term after the first is derived by adding a constant difference from the previous term. This concept is crucial in understanding how salaries increase annually in a structured manner.
For example, the man's salary progression starts at \(30,000\) and increases by \(2,300\) each year. This sequence can be described as:

• Year 1: \(30,000\)
• Year 2: \(30,000 + 2,300 = 32,300\)
• Year 3: \(32,300 + 2,300 = 34,600\)
• ... and so on.

The constant difference, \(d\), is \(2,300\) in this case. Arithmetic progressions provide a clear structure for calculating cumulative income. Understanding this regular pattern simplifies complex calculations and allows us to predict future earnings efficiently.