How Old Is The Youngest Brother? A Mathematical Problem

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#Introduction

In this article, we will delve into a classic mathematical problem involving a family of ten brothers with an age difference between each sibling. We'll break down the problem step by step and guide you through the process of finding the solution. Whether you're a student honing your math skills or simply enjoy the challenge of puzzles, this article is for you. This mathematical problem is a great example of how arithmetic sequences can be applied to solve real-world scenarios. Arithmetic sequences are sequences of numbers in which the difference between any two consecutive terms is constant. In this case, the ages of the brothers form an arithmetic sequence, with a common difference of 2 years.

The problem challenges us to calculate the age of the youngest brother, given that the oldest is 20 years old and each subsequent brother is 2 years younger. It requires a clear understanding of arithmetic sequences and the ability to apply the relevant formula. Let's embark on this mathematical journey together and unravel the age mystery of the youngest brother.

The core of this article lies in solving a specific mathematical problem. In a family of 10 brothers, the oldest brother is 20 years old. Each subsequent brother is 2 years younger than the one before him. The question we aim to answer is: How old is the youngest brother?

This problem requires us to utilize our understanding of arithmetic sequences to determine the age of the youngest brother. We know the age of the oldest brother, the number of brothers, and the age difference between them. By applying the formula for the nth term of an arithmetic sequence, we can systematically calculate the age of the youngest brother.

This problem is not just a mathematical exercise; it also highlights how mathematical concepts can be applied to everyday situations. Understanding arithmetic sequences can help us solve problems related to finance, physics, and various other fields. So, let's delve into the solution and discover the age of the youngest brother.

To effectively solve the brother's age problem, it's crucial to understand the concept of arithmetic sequences. An arithmetic sequence is a series of numbers where the difference between any two consecutive terms remains constant. This constant difference is known as the common difference.

In the context of our problem, the ages of the brothers form an arithmetic sequence. The common difference is 2 years, as each brother is 2 years younger than the previous one. The sequence starts with the oldest brother's age, which is 20, and continues with decreasing ages until we reach the youngest brother.

The general formula for the nth term (an) of an arithmetic sequence is:

an = a1 + (n - 1)d

Where:

  • an is the nth term of the sequence.
  • a1 is the first term of the sequence.
  • n is the position of the term in the sequence.
  • d is the common difference.

In our problem:

  • a1 = 20 (the age of the oldest brother)
  • n = 10 (the total number of brothers)
  • d = -2 (the common difference, which is negative as the ages are decreasing)

By understanding the formula and the values for each variable, we can now proceed to calculate the age of the youngest brother.

Now, let's apply our understanding of arithmetic sequences to solve the problem step by step. Our goal is to find the age of the youngest brother, which is the 10th term in the sequence of brothers' ages.

Step 1: Identify the known values.

  • a1 (the age of the oldest brother) = 20
  • n (the number of brothers) = 10
  • d (the common difference) = -2

Step 2: Apply the formula for the nth term of an arithmetic sequence.

We'll use the formula: an = a1 + (n - 1)d

Step 3: Substitute the known values into the formula.

a10 = 20 + (10 - 1) * (-2)

Step 4: Simplify the equation.

a10 = 20 + (9) * (-2)

a10 = 20 - 18

Step 5: Calculate the result.

a10 = 2

Therefore, the age of the youngest brother is 2 years old.

By following these steps and applying the formula for arithmetic sequences, we have successfully determined the age of the youngest brother in the family.

While we've solved the problem using the arithmetic sequence formula, there are alternative approaches we can explore to gain a deeper understanding and verify our solution.

  1. Listing the ages: We can manually list the ages of all the brothers, starting from the oldest and subtracting 2 years for each subsequent brother. This method can be helpful for visualizing the sequence and confirming our result.

    • Brother 1: 20
    • Brother 2: 18
    • Brother 3: 16
    • Brother 4: 14
    • Brother 5: 12
    • Brother 6: 10
    • Brother 7: 8
    • Brother 8: 6
    • Brother 9: 4
    • Brother 10: 2

    This method clearly shows that the youngest brother is 2 years old.

  2. Working backwards: We can start from the oldest brother's age and subtract 2 years repeatedly until we reach the 10th brother. This approach is similar to listing the ages but can be done without writing down the entire sequence.

    20 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 = 2

    Again, this confirms that the youngest brother is 2 years old.

These alternative approaches not only help verify our solution but also provide different perspectives on the problem. They reinforce the concept of arithmetic sequences and demonstrate how the same problem can be solved using various methods.

Arithmetic sequences are not just abstract mathematical concepts; they have numerous real-world applications in various fields. Understanding these applications can make the concept more relatable and demonstrate the practical value of mathematics.

  1. Finance: Arithmetic sequences can be used to model simple interest calculations. If you deposit a fixed amount of money each month into a savings account, the total amount saved over time will form an arithmetic sequence.

  2. Physics: In physics, arithmetic sequences can be used to describe the motion of objects with constant acceleration. For example, the distance traveled by an object falling under gravity increases in an arithmetic sequence.

  3. Computer Science: Arithmetic sequences are used in various algorithms and data structures. For example, they can be used to generate evenly spaced indices in an array.

  4. Everyday Life: Many everyday situations can be modeled using arithmetic sequences. For example, the number of seats in each row of a theater often forms an arithmetic sequence, with each row having a fixed number of additional seats compared to the previous row.

By recognizing these real-world applications, we can appreciate the importance of arithmetic sequences and their role in solving practical problems.

In conclusion, we have successfully solved the problem of determining the age of the youngest brother in a family of 10, where the oldest is 20 and each subsequent brother is 2 years younger. By applying the formula for arithmetic sequences and exploring alternative approaches, we've confirmed that the youngest brother is 2 years old.

This problem serves as a great example of how mathematical concepts can be used to solve real-world scenarios. Understanding arithmetic sequences not only helps us solve such problems but also provides a foundation for more advanced mathematical concepts.

Furthermore, we've discussed the real-world applications of arithmetic sequences in various fields, highlighting their practical significance. From finance to physics and everyday life, arithmetic sequences play a crucial role in modeling and understanding patterns.

We hope this article has provided a clear and comprehensive explanation of the problem and its solution. Whether you're a student learning about arithmetic sequences or simply enjoy mathematical challenges, we encourage you to continue exploring the fascinating world of mathematics and its applications.

Key Takeaways:

  • The age of the youngest brother is 2 years old.
  • Arithmetic sequences are a powerful tool for solving problems involving constant differences.
  • Arithmetic sequences have numerous real-world applications.

Solving mathematical problems like this one not only enhances our problem-solving skills but also broadens our understanding of the world around us. The ability to apply mathematical concepts to real-life situations is a valuable skill that can benefit us in various aspects of life. So, keep practicing, keep exploring, and keep challenging yourself with new mathematical problems!