/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 63 A man gets a job with a salary o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 63

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.

Short Answer

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

Step by step solution

01

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 \( 2,300 \).
02

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.
03

Substitute the Known Values into the Formula

Substitute \( n = 10 \), \( a_1 = 30,000 \), and \( d = 2,300 \) into the formula: \[ S_{10} = \frac{10}{2} \times (2 \times 30,000 + (10 - 1) \times 2,300) \]
04

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 \( 60,000 + 20,700 = 80,700 \).
05

Solve for the Total Earnings

Now calculate \( S_{10} \): \[ S_{10} = \frac{10}{2} \times 80,700 = 5 \times 80,700 = 403,500 \]
06

Conclusion

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

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(a_{1}, a_{2}\) \(a_{3}, \ldots\) is a geometric sequence with common ratio \(r,\) show that the sequence $$\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \dots$$ is also a geometric sequence, and find the common ratio.

Mike buys a ring for his fiancee by paying \(\$ 30\) a month for one year. If the interest rate is \(10 \%\) per year, compounded monthly, what is the price of the ring?

Simplify using the Binomial Theorem. Show that \((1.01)^{100}>2 . \quad\) [ Hint: Note that \((1.01)^{100}=(1+0.01)^{100},\) and use the Binomial Theorem to show that the sum of the first two terms of the expansion is greater than \(2 .]\)

A couple can afford to make a monthly mortgage payment of \(\$ 650 .\) If the mortgage rate is \(9 \%\) and the couple intends to secure a 30 -year mortgage, how much can they borrow?

Jane agrees to buy a car for a down payment of \(\$ 2000\) and payments of \(\$ 220\) per month for 3 years. If the interest rate is \(8 \%\) per year, compounded monthly, what is the actual purchase price of her car?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.