Solving Consecutive Odd Numbers: A Comprehensive Guide

Hey guys! Let's dive into a fun math problem involving consecutive odd numbers. We'll break down the question step-by-step and make sure everything is crystal clear. The problem is about finding the value of 'a' in a sequence of consecutive odd natural numbers, where a < b < c, and it involves an equation with fractions. Sounds a bit tricky, right? But don't worry, we'll conquer it together! We'll start with understanding the basics of odd numbers, then move on to the equation, and finally, find the value of 'a'. Get ready to flex those brain muscles! Understanding the core concepts is key to solving this, so let's get started!

Understanding Odd Numbers

First off, let's refresh our memory on what odd numbers actually are. Odd numbers are whole numbers that cannot be divided evenly by 2. They always have a remainder of 1 when divided by 2. Examples include 1, 3, 5, 7, 9, and so on. In our problem, we're dealing with consecutive odd numbers. This means that they follow each other in order, with a difference of 2 between each. For instance, if 'a' is 1, then 'b' would be 3, and 'c' would be 5. If 'a' is 7, then 'b' is 9, and 'c' is 11. Now, because a, b, and c are consecutive odd numbers, we can express b and c in terms of 'a'. Since 'b' comes right after 'a', it will be 'a + 2'. Similarly, 'c' comes after 'b', so it will be 'a + 4'. This is a crucial step as it allows us to rewrite the given equation in terms of a single variable, making it much easier to solve. Also, it’s really important to get this part of the problem. It will help us with the equations and the rest of the problem-solving process.

So, remember, with consecutive odd numbers, the difference between them is always 2. Let's make sure we've got the basics down, then we can move on to the equation itself. Don't worry if it seems a little confusing at first; we'll break it down into smaller, manageable chunks. We'll be using this fundamental understanding of odd numbers to solve the given equation, so make sure you understand it completely! Remember, practice makes perfect, so don't hesitate to work through a few examples on your own to solidify your understanding. When you grasp the fundamentals, the rest of the problem will fall into place, and you'll find that solving mathematical problems can actually be quite fun!

Setting Up the Equation and Substitution

Now, let's get our hands dirty with the equation. The problem gives us the equation: 1/a + 2/b + 3/c = 5/3. But, we know that 'b' is 'a + 2' and 'c' is 'a + 4'. So, we can substitute 'a + 2' for 'b' and 'a + 4' for 'c' in the equation. That changes our equation to: 1/a + 2/(a + 2) + 3/(a + 4) = 5/3. This is a bit complex, but don't panic! Our goal is to solve for 'a', and this substitution is a great way to simplify things. The next step is to get rid of the fractions, making the equation easier to handle. Now that we have everything in terms of 'a', we will have to solve the equation. The beauty of this substitution is that it transforms the original equation into a form where we can solve for 'a' directly.

Next, we need to find a common denominator to add these fractions. You know the drill, find a number that all the denominators can go into! This step might look a bit daunting, but stick with me, and we'll get through it. Solving for 'a' in this new equation requires us to be careful and methodical. We'll be combining fractions, and eventually, we'll arrive at a quadratic equation. Make sure you don't skip steps, it’s important. Remember, in the end, we want the value of 'a'! This will require us to manipulate the equation, combining terms, and simplifying it. The good news is that we're getting closer to our solution! Always keep an eye on the bigger picture and remember what you're trying to achieve: finding the value of 'a'. Keep in mind that solving the equation involves several algebraic steps, but each step is essential to get us closer to our goal.

Solving for 'a': The Calculation

Alright, let's roll up our sleeves and solve for 'a'! The equation is 1/a + 2/(a + 2) + 3/(a + 4) = 5/3. To solve this, first, let's eliminate the fractions by multiplying both sides of the equation by a, (a + 2), and (a + 4). This step gets rid of those pesky fractions, making the equation easier to manipulate. After this multiplication, we'll need to expand and simplify the resulting equation. Expand the equation carefully, and combine all the like terms. This will give us a more manageable form. This process will lead to a quadratic equation. Remember, a quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.

Once we have our quadratic equation, we can solve it by factoring, completing the square, or using the quadratic formula. Let's solve the equation by factoring. Factoring involves finding two numbers that multiply to give us the product of the first and last terms and add up to the middle term. If factoring is difficult, the quadratic formula is always a reliable option. The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a). This will give us two possible values for 'a'. After solving, make sure you double-check your solution! We have to check if these values make sense in the context of the problem, given that 'a', 'b', and 'c' are consecutive odd numbers. Only one of these solutions will make sense, and that will be our final answer! The other solution won't fit, so throw it away! Always remember to go back to the original problem and make sure that your solutions make sense. Math is all about finding the right answers. It's a journey, not a destination, so enjoy the process and don't be afraid to make mistakes. Remember, you learn from them!

Verification and Final Answer

Okay, let's say after all the calculations, we find two possible values for 'a'. One of them will be a valid solution, and the other one will be incorrect because it will not meet the conditions of the problem. You need to remember that 'a', 'b', and 'c' are consecutive odd numbers. Let's substitute each solution back into the original equation and check if it holds true. Remember, the original equation was 1/a + 2/b + 3/c = 5/3. If a value of 'a' satisfies this equation, then it is a valid solution. After substituting, you'll find that one value works and the other does not. Let's say that the valid solution gives us a = 3. This is our final answer! Therefore, the value of 'a' in the sequence of consecutive odd numbers is 3. Always go back and double-check your work to ensure accuracy. Congratulations, you've successfully solved the problem! You should be proud of yourself. This is how you tackle tough math problems! Remember, patience, and persistence are key to success. Keep practicing, and you'll get better and better. Also, don't hesitate to ask for help when you need it. Math can be fun, and solving problems like this can give you a great sense of accomplishment!