Finding Geometric Progression Real Numbers Sum And Squares Solution
Hey everyone! Today, we're diving into a cool math problem that involves finding three real numbers that form a geometric progression (G.P.). The twist? We know their sum and the sum of their squares. Let's break it down step by step. This is a common type of problem in mathematics, especially when you're dealing with sequences and series. So, if you're into math or just curious, stick around, and let's solve this together!
Understanding Geometric Progression (G.P.)
Before we jump into the problem, let's quickly recap what a geometric progression is. In simple terms, a G.P. is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, 2, 4, 8, 16... is a G.P. where each term is multiplied by 2 to get the next term.
Now, in our problem, we need to find three numbers, let's call them a/r, a, and ar, where 'a' is the middle term and 'r' is the common ratio. This way of representing numbers in a G.P. makes the calculations a bit easier. Think of 'a' as the heart of our sequence and 'r' as the rhythm that keeps it going. The beauty of using this representation is that it simplifies many calculations, especially when dealing with sums and products. Remember, the goal here is not just to find any three numbers, but three numbers that follow this specific pattern of geometric progression. It's like finding the perfect musical notes that harmonize together in a sequence. Each number has a relationship with the others, dictated by the common ratio, which makes this problem more intriguing than just finding any random set of numbers.
Problem Statement
Okay, let's state the problem clearly. We need to determine three real numbers in G.P. such that their sum is 21/8 and the sum of their squares is 189/64. This means we have two key pieces of information: the total of our three numbers and the total of their squares. These are our clues, and like any good detective, we need to use them to uncover the mystery numbers. The sum condition gives us a linear equation, while the sum of squares gives us a quadratic one. Together, they form a system of equations that we can solve. It's like having two different perspectives on the same situation; each one gives us a slightly different angle, and by combining them, we can get a clear picture. The challenge lies in how we manipulate these equations to isolate our unknowns, 'a' and 'r'. Think of it as a mathematical puzzle, where each equation is a piece, and we need to fit them together to reveal the solution. This is where our algebraic skills come into play, as we'll need to use various techniques to simplify and solve the equations.
Setting up the Equations
Based on the problem, we can set up two equations. Let the three numbers be a/r, a, and ar. Then:
- Sum of the numbers: a/r + a + ar = 21/8
- Sum of the squares: (a/r)² + a² + (ar)² = 189/64
These two equations are the foundation of our solution. The first equation represents the sum of the terms in the G.P., while the second equation represents the sum of the squares of these terms. It's like having a blueprint of the problem, where each equation is a key component. Now, the challenge is to manipulate these equations in a way that allows us to solve for 'a' and 'r'. This involves algebraic techniques such as substitution, elimination, and factorization. The first equation is linear in 'a', while the second equation is quadratic, which adds a layer of complexity. However, with careful manipulation, we can transform these equations into a more manageable form. Think of it as decoding a secret message; each step brings us closer to revealing the values of 'a' and 'r'. The beauty of this approach is that it breaks down a complex problem into smaller, more manageable steps, making the solution process more accessible and less daunting.
Solving the Equations
Now comes the fun part – solving these equations! This is where our algebraic skills get a workout. Let's start by simplifying the first equation. We can factor out 'a' from the left side:
a(1/r + 1 + r) = 21/8
Next, let's simplify the second equation. Notice that it involves squares, which might make things a bit trickier, but we'll handle it:
a²/r² + a² + a²r² = 189/64
We can factor out a² from the left side:
a²(1/r² + 1 + r²) = 189/64
Now, we have two equations that look a bit cleaner. The next step is to find a way to relate these equations. One common strategy is to express one variable in terms of the other and then substitute it into the other equation. This is where the magic of algebra comes into play. Think of it as a puzzle, where you're trying to fit the pieces together. We can solve the first equation for 'a' and then substitute this expression into the second equation. This will give us an equation in terms of 'r' only, which we can then solve. Alternatively, we can square the first equation and compare it with the second equation. This might reveal some hidden relationships and simplify the problem. The key is to be strategic and patient, trying different approaches until we find one that works. It's like exploring a maze, where you might encounter dead ends, but eventually, you'll find the right path.
Let's isolate 'a' from the first equation:
a = (21/8) / (1/r + 1 + r)
Now, let's substitute this expression for 'a' into the second equation. This will give us a single equation in terms of 'r'. It might look a bit intimidating at first, but don't worry, we'll break it down step by step.
[(21/8) / (1/r + 1 + r)]² * (1/r² + 1 + r²) = 189/64
This equation looks complex, but we can simplify it. The goal here is to get rid of the fractions and combine like terms. This might involve multiplying both sides by a common denominator or using algebraic identities to simplify the expressions. The key is to be organized and methodical, keeping track of each step. It's like untangling a knot; you need to work carefully and patiently to avoid making it worse. Once we've simplified the equation, we should end up with a polynomial equation in 'r'. This might be a quadratic, cubic, or even a higher-degree polynomial. To solve it, we can use various techniques such as factoring, the quadratic formula, or numerical methods. The specific method will depend on the degree of the polynomial and the coefficients involved. Remember, the goal is to find the values of 'r' that satisfy the equation, as these values will help us determine the terms of the geometric progression.
After simplifying and solving for 'r', we get two possible values for r: r = 2 and r = 1/2. These are the common ratios that define our geometric progressions. Now that we have these values, we can plug them back into our equation for 'a' to find the corresponding values of 'a'. It's like finding the missing pieces of a puzzle; once you have the common ratio, you can unlock the other terms in the sequence. For each value of 'r', we'll have a different value of 'a', which will give us a different set of numbers in the G.P. This means we might have multiple solutions to our problem. It's important to check each solution to make sure it satisfies the original conditions of the problem. This is like verifying your answer in a math test; you want to be sure that your solution is correct and makes sense in the context of the problem. Remember, math is not just about finding the answer, it's also about understanding why the answer is correct.
Finding the Numbers
Now that we have the values for 'r', we can find the corresponding values for 'a'.
For r = 2: a = (21/8) / (1/2 + 1 + 2) = (21/8) / (7/2) = 3/4
For r = 1/2: a = (21/8) / (2 + 1 + 1/2) = (21/8) / (7/2) = 3/4
Interestingly, we get the same value for 'a' in both cases, which is 3/4. This simplifies our task, as we only have one value of 'a' to work with. Now, we can plug in the values of 'a' and 'r' into our expressions for the three numbers in the G.P.: a/r, a, and ar. This will give us the specific numbers that satisfy the conditions of the problem. It's like putting the finishing touches on a painting; you're taking the basic elements and combining them to create the final product. For each pair of 'a' and 'r' values, we'll get a different set of numbers in the G.P. These numbers will form a sequence where each term is related to the others by the common ratio. It's important to check that these numbers indeed form a geometric progression and that their sum and the sum of their squares match the values given in the problem. This is like testing your recipe to make sure the flavors are balanced and the dish tastes delicious.
Now, let's calculate the numbers for each case:
Case 1: r = 2, a = 3/4
- a/r = (3/4) / 2 = 3/8
- a = 3/4
- ar = (3/4) * 2 = 3/2
So, the numbers are 3/8, 3/4, and 3/2.
Case 2: r = 1/2, a = 3/4
- a/r = (3/4) / (1/2) = 3/2
- a = 3/4
- ar = (3/4) * (1/2) = 3/8
So, the numbers are 3/2, 3/4, and 3/8. Notice that this is just the reverse of the first sequence. This is a common occurrence in G.P. problems, where the sequence can be reversed by taking the reciprocal of the common ratio. It's like looking at the same sequence from a different perspective; the numbers are the same, but their order is reversed. This can be a useful observation when solving G.P. problems, as it can help you find multiple solutions or simplify your calculations.
Final Answer
Therefore, the three real numbers in geometric progression are 3/8, 3/4, and 3/2 (or 3/2, 3/4, and 3/8). These numbers satisfy both conditions of the problem: their sum is 21/8, and the sum of their squares is 189/64. It's like completing a puzzle; you've found the missing pieces and put them together to form a coherent picture. The journey to find these numbers involved understanding geometric progressions, setting up equations, solving those equations, and finally, calculating the numbers themselves. Each step was crucial, and by following a systematic approach, we were able to arrive at the solution. Math is like a journey of discovery, where each problem is a new challenge, and the solution is the reward. So, keep exploring, keep questioning, and keep solving!
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Finding Geometric Progression Real Numbers Sum and Squares Solution
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Find three real numbers in geometric progression such that their sum is 21/8 and the sum of their squares is 189/64.
