If (a1, A2, A3) Is A Geometric Progression With A Common Ratio Of 3/2 And A1 + A2 + A3 = 95, Then What Is The Value Of A1 – A2 + A3?

Let's dive into the heart of geometric progressions (G.P.) with this intriguing problem. Geometric progressions, at their core, are sequences where each term is derived by multiplying the preceding term by a constant factor, aptly termed the 'common ratio.' Understanding this fundamental property is paramount to unlocking the solution. In this article, we will dissect the problem step by step, ensuring clarity and comprehension every step of the way. The question before us presents a G.P. denoted as (a1, a2, a3) with a common ratio of 3/2. A crucial piece of information is that the sum of these terms, a1 + a2 + a3, equals 95. The challenge lies in determining the value of a slightly modified expression: a1 – a2 + a3. To successfully navigate this problem, we will leverage the defining characteristic of geometric progressions – the common ratio – and strategically manipulate the given equation to isolate and determine the individual terms. The allure of this problem lies not just in finding the numerical answer, but in the journey of applying mathematical principles to unravel the unknown. As we embark on this journey, we will not only solve for the desired expression but also reinforce our understanding of geometric progressions and their applications. So, let's put on our mathematical thinking caps and embark on this quest to decipher the solution.

Problem Restatement: Decoding the Essence of the Question

Before we plunge into the calculations, let's make absolutely sure we grasp the problem's essence. We are presented with a geometric progression, a sequence of numbers where each term is found by multiplying the previous one by a constant factor. In this specific instance, our G.P. is (a1, a2, a3), and the common ratio, the linchpin connecting consecutive terms, is 3/2. This means that a2 is equal to a1 multiplied by 3/2, and similarly, a3 is a2 multiplied by 3/2 (or a1 multiplied by (3/2)^2). The problem further reveals that the sum of these three terms – a1, a2, and a3 – amounts to 95. This provides us with a concrete equation to work with. The ultimate question posed is: what is the value of the expression a1 – a2 + a3? Notice the subtle but significant change from the given sum; the second term, a2, is now subtracted instead of added. This seemingly minor alteration necessitates a careful and methodical approach to the solution. Our objective is clear: we need to find the individual values of a1, a2, and a3, or, perhaps more efficiently, find a way to directly calculate a1 – a2 + a3 without explicitly determining each term. To achieve this, we will harness the power of the common ratio and the given sum equation, strategically manipulating them to unveil the desired answer. With a clear understanding of the problem's requirements, we are now poised to embark on the solution process, armed with the right tools and a focused approach.

Solution: Unraveling the Geometric Progression

Now, let's roll up our sleeves and dive into the solution. Remember, we have a geometric progression (a1, a2, a3) with a common ratio of 3/2, and we know that a1 + a2 + a3 = 95. Our mission is to find the value of a1 – a2 + a3. The key to unlocking this problem lies in expressing a2 and a3 in terms of a1, leveraging the common ratio. Since the common ratio is 3/2, we can write: a2 = a1 * (3/2) and a3 = a2 * (3/2) = a1 * (3/2) * (3/2) = a1 * (9/4). Now, we can substitute these expressions for a2 and a3 into the given sum equation: a1 + a1 * (3/2) + a1 * (9/4) = 95. This equation now involves only one unknown, a1, which we can solve for. To simplify the equation, let's find a common denominator for the fractions, which is 4. Multiplying each term by the appropriate factor, we get: (4/4) * a1 + (6/4) * a1 + (9/4) * a1 = 95. Combining the terms, we have: (4a1 + 6a1 + 9a1) / 4 = 95, which simplifies to 19a1 / 4 = 95. To isolate a1, we multiply both sides of the equation by 4: 19a1 = 95 * 4 = 380. Finally, we divide both sides by 19: a1 = 380 / 19 = 20. Now that we have found a1, we can easily determine a2 and a3: a2 = a1 * (3/2) = 20 * (3/2) = 30 and a3 = a1 * (9/4) = 20 * (9/4) = 45. With the values of a1, a2, and a3 in hand, we can now calculate the desired expression: a1 – a2 + a3 = 20 – 30 + 45 = 35. Therefore, the value of a1 – a2 + a3 is 35. This methodical approach, breaking down the problem into manageable steps and utilizing the fundamental properties of geometric progressions, has led us to the solution.

Answer: The Final Verdict

After our meticulous journey through the problem, we have arrived at the final answer. We successfully navigated the intricacies of the geometric progression, leveraging the common ratio and the given sum to determine the values of the individual terms. By expressing a2 and a3 in terms of a1, we were able to construct an equation that allowed us to solve for a1. Once we had a1, the rest fell into place, and we readily calculated a2 and a3. Finally, we plugged these values into the target expression, a1 – a2 + a3, and the result emerged: 35. Therefore, the correct answer is C) 35. This problem serves as a testament to the power of understanding fundamental mathematical principles and applying them strategically. The geometric progression, with its inherent pattern and predictable relationships between terms, can be conquered with the right approach. By breaking down the problem into smaller, manageable steps, we were able to unravel the complexities and arrive at the solution with confidence. The journey through this problem has not only provided us with a numerical answer but has also reinforced our understanding of geometric progressions and the art of problem-solving.