Finding Consecutive Natural Numbers With Sum Of Squares
This article explores the fascinating realm of number theory, specifically focusing on problems involving consecutive natural numbers and the sums of their squares. We'll delve into two distinct yet related challenges:
- Finding two consecutive natural numbers whose squares add up to 545.
- Determining three consecutive natural numbers where the sum of their squares equals 149.
These problems, while seemingly simple, offer a great way to understand algebraic techniques and problem-solving strategies within mathematics. Let's embark on this numerical journey and uncover the solutions step by step.
Problem 1: Two Consecutive Natural Numbers Summing to 545
Setting up the Equation
In this section, our main keyword is consecutive natural numbers. To solve the first problem, we need to translate the word problem into a mathematical equation. Let's denote the first natural number as 'n'. Since we are looking for consecutive natural numbers, the next number will be 'n + 1'. The problem states that the sum of the squares of these two numbers is 545. Therefore, we can write the equation as follows:
n² + (n + 1)² = 545
This equation is a quadratic equation, and solving it will give us the value of 'n'. Once we find 'n', we can determine the two consecutive natural numbers. The beauty of algebra lies in its ability to represent real-world problems in a symbolic form, making them easier to manipulate and solve. This equation is a perfect example of how we can use algebraic expressions to model numerical relationships. By understanding the problem's conditions and translating them into an equation, we pave the way for finding the solution. Now, let's proceed to solve this equation and unveil the two consecutive natural numbers that satisfy the given condition. The equation, n² + (n + 1)² = 545, encapsulates the heart of our problem, and the following steps will guide us towards the answer. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. By carefully setting up the equation and applying the rules of algebra, we can confidently arrive at the solution. This process not only helps us find the answer but also enhances our problem-solving skills, making us better mathematicians in the long run.
Solving the Quadratic Equation
The main focus here is on solving the equation. Now, let's solve the quadratic equation we derived in the previous section: n² + (n + 1)² = 545. First, we need to expand the equation and simplify it. Expanding (n + 1)² gives us n² + 2n + 1. So, the equation becomes:
n² + n² + 2n + 1 = 545
Combining like terms, we get:
2n² + 2n + 1 = 545
Next, we subtract 545 from both sides to set the equation to zero:
2n² + 2n - 544 = 0
To simplify the equation further, we can divide all terms by 2:
n² + n - 272 = 0
Now, we have a simpler quadratic equation to solve. We can solve this equation by factoring, completing the square, or using the quadratic formula. Factoring might be tricky in this case, so let's use the quadratic formula. The quadratic formula is given by:
n = (-b ± √(b² - 4ac)) / (2a)
In our equation, a = 1, b = 1, and c = -272. Plugging these values into the formula, we get:
n = (-1 ± √(1² - 4 * 1 * -272)) / (2 * 1) n = (-1 ± √(1 + 1088)) / 2 n = (-1 ± √1089) / 2 n = (-1 ± 33) / 2
This gives us two possible solutions for n:
n = (-1 + 33) / 2 = 32 / 2 = 16 n = (-1 - 33) / 2 = -34 / 2 = -17
Since we are looking for natural numbers, we discard the negative solution. Therefore, n = 16.
Finding the Consecutive Numbers
The crucial step here is to identify the numbers. We found that the first natural number, n, is 16. The next consecutive natural number is n + 1, which is 16 + 1 = 17. So, the two consecutive natural numbers are 16 and 17. To verify our solution, we can check if the sum of their squares is indeed 545:
16² + 17² = 256 + 289 = 545
Our solution is correct. Therefore, the two consecutive natural numbers whose squares have a sum of 545 are 16 and 17. This problem demonstrates how we can use algebraic equations to represent real-world scenarios and solve them systematically. The process of setting up the equation, simplifying it, and then solving for the unknown variable is a fundamental skill in mathematics. Moreover, understanding the nature of the solutions, such as discarding negative values when looking for natural numbers, is equally important. This problem not only provides us with a specific answer but also reinforces our understanding of mathematical concepts and problem-solving techniques. The elegance of mathematics lies in its ability to provide precise solutions to seemingly complex problems, and this example perfectly illustrates that.
Problem 2: Three Consecutive Natural Numbers Summing to 149
Setting up the Equation
Now, let's tackle the second problem, which involves finding three consecutive natural numbers whose squares sum up to 149. Here, the main keyword is three consecutive natural numbers. Similar to the previous problem, we need to translate the word problem into a mathematical equation. Let's denote the first natural number as 'n'. Then the next two consecutive natural numbers will be 'n + 1' and 'n + 2'. The problem states that the sum of the squares of these three numbers is 149. Therefore, we can write the equation as follows:
n² + (n + 1)² + (n + 2)² = 149
This equation is also a quadratic equation, and solving it will give us the value of 'n'. Once we find 'n', we can determine the three consecutive natural numbers. This equation beautifully represents the relationship between three consecutive natural numbers and the sum of their squares. By setting up the equation correctly, we have taken the first crucial step towards solving the problem. The algebraic representation allows us to manipulate the equation and apply mathematical techniques to find the unknown values. Understanding the problem's conditions and translating them into an equation is a key skill in mathematics, and this problem provides a great opportunity to practice that skill. Now, let's move forward and solve this equation to uncover the three consecutive natural numbers that satisfy the given condition. The equation, n² + (n + 1)² + (n + 2)² = 149, is the foundation of our solution, and the subsequent steps will guide us towards the answer. Remember, the essence of problem-solving lies in breaking down complex problems into simpler steps. By carefully setting up the equation and applying algebraic principles, we can confidently find the solution and enhance our mathematical abilities.
Solving the Quadratic Equation
The focus of this section is the solution. To solve the equation n² + (n + 1)² + (n + 2)² = 149, we first need to expand and simplify it. Expanding (n + 1)² gives us n² + 2n + 1, and expanding (n + 2)² gives us n² + 4n + 4. So, the equation becomes:
n² + (n² + 2n + 1) + (n² + 4n + 4) = 149
Combining like terms, we get:
3n² + 6n + 5 = 149
Next, we subtract 149 from both sides to set the equation to zero:
3n² + 6n - 144 = 0
To simplify the equation further, we can divide all terms by 3:
n² + 2n - 48 = 0
Now, we have a simpler quadratic equation to solve. We can solve this equation by factoring, completing the square, or using the quadratic formula. In this case, factoring seems to be the easiest method. We are looking for two numbers that multiply to -48 and add up to 2. Those numbers are 8 and -6. So, we can factor the equation as follows:
(n + 8)(n - 6) = 0
This gives us two possible solutions for n:
n + 8 = 0 => n = -8 n - 6 = 0 => n = 6
Since we are looking for natural numbers, we discard the negative solution. Therefore, n = 6.
Finding the Consecutive Numbers
The important step is to determine the numbers. We found that the first natural number, n, is 6. The next two consecutive natural numbers are n + 1 = 6 + 1 = 7 and n + 2 = 6 + 2 = 8. So, the three consecutive natural numbers are 6, 7, and 8. To verify our solution, we can check if the sum of their squares is indeed 149:
6² + 7² + 8² = 36 + 49 + 64 = 149
Our solution is correct. Therefore, the three consecutive natural numbers whose squares have a sum of 149 are 6, 7, and 8. This problem further illustrates the power of algebraic equations in solving real-world problems. The process of setting up the equation, simplifying it, and then solving for the unknown variable is a fundamental skill in mathematics. Moreover, understanding the nature of the solutions and applying techniques like factoring to solve quadratic equations are crucial. This problem not only provides us with a specific answer but also strengthens our understanding of mathematical concepts and problem-solving methodologies. The beauty of mathematics lies in its ability to provide precise and elegant solutions to seemingly complex problems, and this example perfectly showcases that.
In summary, we have successfully solved two problems involving consecutive natural numbers and the sums of their squares. The first problem led us to the consecutive numbers 16 and 17, while the second problem revealed the sequence 6, 7, and 8. These exercises highlight the application of algebraic techniques in solving number theory problems. The process of translating word problems into mathematical equations, simplifying those equations, and then solving for the unknowns is a fundamental skill in mathematics. Moreover, these problems underscore the importance of understanding the nature of numbers and applying appropriate problem-solving strategies. The beauty of mathematics lies in its ability to provide precise and logical solutions to complex problems, and these examples serve as a testament to that elegance. By mastering these techniques, we can approach a wide range of mathematical challenges with confidence and clarity. These problems not only provide us with specific answers but also enhance our overall mathematical reasoning and problem-solving abilities.
