Solving Geometric Progression And Variation Problems A Step-by-Step Guide
Hey guys! Ever get tangled up in the world of geometric progressions and variations? Don't sweat it! We're about to break down a tricky problem and make it super clear. We'll tackle a geometric progression (G.P.) question and a variation problem step-by-step. So, grab your thinking caps, and let's dive in!
Geometric Progression (G.P.) Decoded
Understanding Geometric Progressions
Geometric Progressions (G.P.) are sequences where each term is found by multiplying the previous term by a constant factor. This constant factor is called the common ratio (r). To fully grasp G.P., imagine a series of numbers like 2, 6, 18, 54, and so on. Notice how each number is three times the previous one? That's a geometric progression in action, with a common ratio of 3. Understanding G.P. isn't just about recognizing patterns; it's crucial for tackling problems involving exponential growth or decay in various real-world scenarios, from financial investments to population dynamics. The foundational formula for the nth term of a G.P. is given by an = arn-1, where a is the first term and r is the common ratio. This formula is your go-to tool for finding any term in the sequence, provided you know the first term and the common ratio. But what if you don't know these key pieces? That's where problem-solving skills come into play. Often, you'll be given certain terms in the sequence and asked to find others or the initial values. This requires setting up equations and using algebraic manipulation to uncover the unknowns. For example, if you're given the third and seventh terms of a G.P., you can set up two equations using the formula an = arn-1 and solve them simultaneously to find a and r. Mastering these techniques not only helps in solving textbook problems but also sharpens your analytical thinking, preparing you for more complex mathematical challenges.
Problem Breakdown: Finding the First Term
In this specific problem, we're given that the third term of a G.P. is 24 and the seventh term is 4 20/27. Our mission? To find the first term. Let's break it down. The main challenge here is that we don't know the first term (a) or the common ratio (r). But, we do have enough information to create a system of equations. Remember that magic formula, an = arn-1? We'll use it twice, once for the third term and once for the seventh term. For the third term (n=3), we have a3 = ar3-1 = ar2. We know this equals 24, so our first equation is ar2 = 24. Next, for the seventh term (n=7), we have a7 = ar7-1 = ar6. This equals 4 20/27, which we can convert to an improper fraction: (4 * 27 + 20) / 27 = 128/27. So, our second equation is ar6 = 128/27. Now we have two equations with two unknowns. The trick is to solve these simultaneously. A common method is to divide the second equation by the first. This eliminates a and leaves us with an equation solely in terms of r. By solving for r, we can then substitute it back into either equation to find a. This process highlights a powerful problem-solving strategy: breaking down a complex problem into smaller, manageable steps. By identifying the key information, setting up appropriate equations, and using algebraic techniques, we can systematically unravel the solution.
Step-by-Step Solution
Let’s dive into the nitty-gritty and solve this geometric progression puzzle step-by-step, guys! We have two key pieces of information: the third term (a3) is 24, and the seventh term (a7) is 4 20/27 (which we converted to 128/27). Remember the formula for the nth term of a G.P.: an = arn-1, where a is the first term and r is the common ratio. First, let's express the given information as equations. For the third term, we have a3 = ar2 = 24. This is our Equation 1. For the seventh term, we have a7 = ar6 = 128/27. This is our Equation 2. The next step is to eliminate one of the variables. The easiest way to do this is to divide Equation 2 by Equation 1. This gives us (ar6) / (ar2) = (128/27) / 24. Simplifying this, the a cancels out, and we're left with r4 = 128 / (27 * 24). Further simplification gives us r4 = 128 / 648, which reduces to r4 = 64 / 324, and further to r4 = 16/81. To find r, we take the fourth root of both sides: r = ± √(2/3). Now, we have two possible values for r: 2/3 and -2/3. We'll consider both cases. To find a, we substitute each value of r back into Equation 1 (ar2 = 24). Let's start with r = 2/3. Substituting, we get a(2/3)2 = 24, which simplifies to a(4/9) = 24. Solving for a, we get a = 24 * (9/4) = 54. So, when r = 2/3, a = 54. Now, let's consider r = -2/3. Substituting into Equation 1, we get a(-2/3)2 = 24, which is the same as a(4/9) = 24. Again, solving for a, we get a = 54. Interestingly, the value of a is the same for both values of r. Therefore, the first term of the G.P. is 54.
Variations: Direct and Inverse
Grasping Direct and Inverse Variations
Variations describe how quantities change in relation to each other. Direct variation means that as one quantity increases, the other increases proportionally, and vice versa. Think of it like this: the more hours you work, the more money you earn. This relationship can be expressed as y = kx, where y and x are the quantities that vary directly, and k is the constant of variation. This constant k is the magic number that ties the two quantities together, showing how much y changes for every unit change in x. Understanding direct variation is crucial in everyday scenarios, such as calculating the cost of multiple items based on the price of one, or determining the distance traveled based on speed and time. On the flip side, inverse variation means that as one quantity increases, the other decreases, and vice versa. Imagine the relationship between speed and time for a fixed distance: the faster you go, the less time it takes. This is expressed as y = k/x, where y and x vary inversely, and k is, again, the constant of variation. This constant represents the product of the two varying quantities, which remains constant. Inverse variation pops up in various contexts, such as understanding the relationship between the number of workers and the time it takes to complete a job, or the pressure and volume of a gas at a constant temperature. Now, things get interesting when we mix these variations. A quantity can vary directly with one variable and inversely with another (or even the square or cube of another!). For instance, the gravitational force between two objects varies directly with the product of their masses and inversely with the square of the distance between them. To tackle these combined variations, we combine the individual relationships into a single equation. If y varies directly as x and inversely as the square of z, the combined variation is expressed as y = kx/z2. Mastering these concepts allows you to model and predict how different variables interact, making it a powerful tool in fields ranging from physics and engineering to economics and data analysis.
Problem Setup: Combining Variations
Let's tackle a variation problem! We're told that y varies directly as x and inversely as the square of z. This is a classic combined variation scenario. The key phrase here is