Solving For X And Y: Geometric And Arithmetic Progressions

Hey guys! Let's dive into a cool math problem. We're gonna figure out the values of x and y. We've got two sequences here: one's a Geometric Progression (PG), and the other's an Arithmetic Progression (PA). The catch? We need to use what we know about these sequences to find our answers. This problem is like a little puzzle, and we'll break it down step-by-step to crack it. Ready to get started?

Understanding the Problem: Unpacking Geometric and Arithmetic Progressions

Okay, so the problem throws two sequences at us: (1, x, y) and (x, y, 6). The first one, (1, x, y), is a Geometric Progression (PG). That means each term is found by multiplying the previous term by a constant value, which we call the 'common ratio'. Importantly, this common ratio is positive, which will be key later. The second sequence, (x, y, 6), is an Arithmetic Progression (PA). This one's different: in an AP, you add a constant value to get from one term to the next – this is the 'common difference'.

Now, let's get into what these mean for us. In a PG, the ratio between consecutive terms is always the same. So, x/1 = y/ x. We can simplify this to x² = y. That's our first critical equation! It tells us how x and y relate to each other in the PG.

For the AP, the difference between consecutive terms is constant. This means y - x = 6 - y. If we rearrange this, we get 2y = x + 6. This is our second important equation. Notice how we've gone from the information in the problem to two equations, that gives us a system of equations, ready to be solved. Let's make sure we've understood everything so far. We are looking for values of x and y.

We know that (1, x, y) is a PG with a positive ratio, and (x, y, 6) is an AP. Now we're going to solve this using systems of equations. To solve this, we can substitute our equation and find the answers to x and y. Now that we've got the basics down, we can start to bring our knowledge of the math concepts together to solve this problem. These are two different types of mathematical sequences that follow specific rules.

We're dealing with a PG, where a common ratio is applied. With a PA, it's about a common difference. These little details are super important for solving the problem. The question wants us to figure out the values of x and y. Don't worry, we're going to make sure that we understand the core concepts. The fact that the ratio is positive is a sneaky detail that can influence our answer later on. The most important thing here is to understand the different progressions, that will help solve this problem. Let's dive into the core concepts.

Setting up the Equations: From Sequences to Algebra

Alright, time to get our hands dirty with some algebra! We've already got our two crucial equations, derived directly from the definitions of PG and AP. Let's write them down again for clarity:

1. x² = y (from the PG)
2. 2y = x + 6 (from the AP)

Our goal now is to solve these equations simultaneously to find the values of x and y. The easiest way to solve this system is by substitution. Since we know y = x² (from equation 1), we can plug this value of y into equation 2. Doing so gives us: 2(x²) = x + 6. Now, let's rearrange this equation to get a standard quadratic equation: 2x² - x - 6 = 0.

See? It's all starting to come together! We've transformed our sequence problem into a nice, solvable quadratic equation. So we can use the quadratic formula to solve for x. The quadratic formula is x = (-b ± √( b² - 4ac )) / (2a). In our equation (2x² - x - 6 = 0), a = 2, b = -1, and c = -6.

Plugging these values into the quadratic formula, we get: x = (1 ± √(1 - 4 * 2 * -6)) / 4. This simplifies to x = (1 ± √49) / 4. Therefore, x = (1 ± 7) / 4. So, we have two possible values for x: x = 2 or x = -3/2. Now we will use the results of this equation to solve the other questions.

Solving for x and y: The Math Behind the Solution

Okay, we've got two possible values for x: 2 and -3/2. Now, we need to find the corresponding y values for each. Remember our first equation from the PG: x² = y.

If x = 2, then y = 2² = 4. So, one possible solution is x = 2 and y = 4. Let's check if this solution fits our criteria. The PG would be (1, 2, 4), with a common ratio of 2 (positive, as required). The AP would be (2, 4, 6), with a common difference of 2. So, this solution works!

Now, let's check what happens if x = -3/2. Then, y = (-3/2)² = 9/4. Our PG would be (1, -3/2, 9/4), which has a common ratio of -3/2. However, the problem states that the PG has a positive ratio. This means this solution doesn't work. Thus, the only valid solution is x = 2 and y = 4.

Let's review our understanding. We began with sequences and ended up with a simple quadratic equation. Solving that equation, we had to pick the only one that was valid. Make sure to keep in mind the details of the problem.

The Final Answer and a Recap: Putting It All Together

So, after all the calculations and checks, we've arrived at our final answer: x = 2 and y = 4. The PG is (1, 2, 4), and the AP is (2, 4, 6). We carefully analyzed the problem, set up our equations based on the definitions of PG and AP, solved the equations, and then validated our results, ensuring they met the specific conditions of the problem. That's the key to solving this kind of problem!

This problem nicely illustrates the interconnectedness of different math concepts. We used the properties of sequences to create algebraic equations, and then employed our knowledge of solving equations to find the solution. And, by making sure our solution met all the conditions (like the positive common ratio), we were able to arrive at the correct answer.

So guys, that wraps up our problem-solving session! Hopefully, this has been helpful. Keep practicing and applying these concepts. You'll get better with each problem you tackle! Remember that the most important thing is understanding the rules of progressions, and setting up the equations in a correct way, the math will follow. Keep practicing and keep learning! We have solved the equation and the work is done. Well done!