Solving Geometric Progression Problems Step-by-Step

Hey guys! Today, we're diving deep into the fascinating world of geometric progressions (GPs). We'll tackle some interesting problems, breaking them down step by step so you can master these concepts. So, grab your thinking caps, and let's get started!

Problem 48 Finding the Middle Ground in a 7-Term GP

In geometric progressions, understanding the relationship between terms is super important. Geometric progressions (GPs) are sequences where each term is found by multiplying the previous term by a constant factor, known as the common ratio. This constant ratio is the key to unlocking many problems related to GPs. When we're faced with a GP problem, especially one with a limited number of terms, we can use the properties of the GP to find missing terms. Consider the problem where we need to determine a specific term within a GP, given only the first and last terms and the total number of terms. This requires a blend of algebraic manipulation and a solid grasp of GP principles. So, let's break it down, shall we? The problem states: The extremes of a GP of 7 terms are 4 and 256. Determine the fourth term. In this scenario, we know the first term ($a_1$) is 4, and the seventh term ($a_7$) is 256. We need to find the fourth term ($a_4$). To do this, we first need to find the common ratio (r). Remember, in a GP, the nth term ($a_n$) can be expressed as: $a_n = a_1 * r^(n-1)$. So, for the 7th term, we have: $a_7 = a_1 * r^(7-1)$, which simplifies to $256 = 4 * r^6$. Now, let's solve for r. Dividing both sides by 4, we get: $r^6 = 64$. Taking the sixth root of both sides, we find that $r = 2$ (we consider the positive root since it's the most common in these types of problems). Now that we have the common ratio, we can find the fourth term ($a_4$). Using the formula again: $a_4 = a_1 * r^(4-1)$, which gives us $a_4 = 4 * 2^3$. Calculating this, we get: $a_4 = 4 * 8 = 32$. Therefore, the fourth term of the GP is 32. This problem highlights the importance of understanding the fundamental formula of a GP and how to use it to find missing terms. By identifying the given information and applying the formula correctly, we can solve for any term in the sequence. Remember, the key is to break down the problem into smaller, manageable steps and to utilize the properties of GPs effectively.

Problem 49 Unraveling the Ratio of a GP

Finding the common ratio in a GP when given certain relationships between terms can feel like detective work. But don't worry, guys, we've got this! The common ratio is the backbone of a geometric progression. It’s the value that, when multiplied by a term, gives you the next term in the sequence. In this problem, we're not given the terms directly, but rather relationships between them. This requires us to use our algebraic skills to unravel the value of the common ratio. So, let's put on our detective hats and see how we can solve this. The problem statement is: Determine the ratio of a GP such that: $a_1 + a_4 = 28$ and $a_2 + a_5 = 84$. Here, we have two equations involving terms of the GP. Our goal is to find the common ratio (r). Let's express the terms in terms of $a_1$ and r. We know that: $a_2 = a_1 * r$, $a_4 = a_1 * r^3$, and $a_5 = a_1 * r^4$. Now, we can rewrite the given equations: 1. $a_1 + a_1 * r^3 = 28$ 2. $a_1 * r + a_1 * r^4 = 84$ Let's factor out $a_1$ from both equations: 1. $a_1(1 + r^3) = 28$ 2. $a_1 * r(1 + r^3) = 84$ Now, we can divide the second equation by the first equation to eliminate $a_1$: $[a_1 * r(1 + r^3)] / [a_1(1 + r^3)] = 84 / 28$. This simplifies to: $r = 3$. Therefore, the common ratio of the GP is 3. This problem demonstrates a powerful technique in solving GP problems: expressing terms in relation to the first term and the common ratio. By doing this, we can create equations that allow us to solve for the unknowns. The division trick we used here is also a common method for simplifying equations in GPs. Remember, when faced with similar problems, try to express the terms in a way that allows you to eliminate variables and solve for the desired quantity. The beauty of GPs lies in their structured nature, and understanding how to manipulate the terms is key to mastering these problems.

Problem 50 Cracking the General Term of a GP

Figuring out the general term (T.G.) of a GP is like finding the master key to the entire sequence. It allows us to calculate any term without having to compute all the preceding ones. The general term of a geometric progression is a formula that expresses any term of the sequence based on its position in the sequence. It's like having a roadmap to the GP, where you can jump to any term you want without having to go through the entire sequence step by step. This problem gives us some clues in the form of relationships between terms, and we need to piece them together to find the T.G. So, let's see how we can unlock this master key. The problem states: Write the T.G. of a GP, knowing that $a_2 + a_4 + a_6 = 10$ and $a_3 + a_5 + a_7 = 30$. Our goal is to find the general term, which is usually expressed in the form $a_n = a_1 * r^(n-1)$, where $a_1$ is the first term and r is the common ratio. We need to find $a_1$ and r. Let's express the given terms in terms of $a_1$ and r: $a_2 = a_1 * r$, $a_3 = a_1 * r^2$, $a_4 = a_1 * r^3$, $a_5 = a_1 * r^4$, $a_6 = a_1 * r^5$, $a_7 = a_1 * r^6$. Now, let's rewrite the given equations: 1. $a_1 * r + a_1 * r^3 + a_1 * r^5 = 10$ 2. $a_1 * r^2 + a_1 * r^4 + a_1 * r^6 = 30$ Factor out common terms: 1. $a_1 * r(1 + r^2 + r^4) = 10$ 2. $a_1 * r^2(1 + r^2 + r^4) = 30$ Now, divide the second equation by the first equation: $[a_1 * r^2(1 + r^2 + r^4)] / [a_1 * r(1 + r^2 + r^4)] = 30 / 10$. This simplifies to: $r = 3$. Now that we have r, we can substitute it back into the first equation to find $a_1$: $a_1 * 3(1 + 3^2 + 3^4) = 10$, which simplifies to $a_1 * 3(1 + 9 + 81) = 10$, further simplifying to $a_1 * 3(91) = 10$. Thus, $a_1 = 10 / (3 * 91) = 10 / 273$. Now we have both $a_1$ and r, so we can write the general term: $a_n = (10 / 273) * 3^(n-1)$. This is the general term of the GP. This problem showcases how to use relationships between terms to find the general formula of a GP. By expressing the terms using $a_1$ and r, we can create equations that lead us to the values we need. The ability to manipulate these equations and solve for the unknowns is a crucial skill in working with GPs. Remember, the general term is a powerful tool that allows you to understand and predict the behavior of the entire sequence.

Problem 51 Calculating...

(The problem statement is incomplete in the original prompt. To provide a complete solution, we need the full problem statement. However, I can offer a general approach to solving GP problems. Let's assume the problem asks us to calculate a specific term or sum of a GP given some information.)

When calculating terms or sums in geometric progressions, it's essential to identify what information you have and what you need to find. Whether you're calculating a specific term or the sum of a series, having a strategic approach can make the process much smoother. Geometric progressions have specific formulas that govern their behavior, and knowing when and how to apply these formulas is key to success. Let's outline a general approach and then illustrate it with a hypothetical example. First, identify the given information: This might include the first term ($a_1$), the common ratio (r), the number of terms (n), or the sum of the series (S_n). Sometimes, you might be given relationships between terms, similar to the problems we discussed earlier. Next, determine what you need to find: Are you looking for a specific term ($a_n$), the sum of the first n terms (S_n), or something else? Choose the appropriate formula: If you need to find a specific term, use the formula: $a_n = a_1 * r^(n-1)$. If you need to find the sum of the first n terms, use one of the following formulas:

• If $r ≠ 1$: $S_n = a_1 * (1 - r^n) / (1 - r)$
• If $r = 1$: $S_n = n * a_1$ Substitute the given values into the formula and solve: Be careful with your calculations and make sure you're using the correct order of operations. Let's illustrate this with a hypothetical example: Suppose we have a GP where the first term ($a_1$) is 2, the common ratio (r) is 3, and we want to find the sum of the first 5 terms (S_5). 1. We have: $a_1 = 2$, $r = 3$, and $n = 5$. 2. We need to find: $S_5$. 3. Choose the appropriate formula: Since $r ≠ 1$, we use: $S_n = a_1 * (1 - r^n) / (1 - r)$. 4. Substitute the values: $S_5 = 2 * (1 - 3^5) / (1 - 3)$. 5. Solve: $S_5 = 2 * (1 - 243) / (-2) = 2 * (-242) / (-2) = 242$. So, the sum of the first 5 terms is 242. This general approach can be applied to a wide range of GP problems. Remember to carefully identify the given information, determine what you need to find, choose the appropriate formula, and substitute the values correctly. With practice, you'll become more comfortable with these calculations and be able to solve GP problems with confidence. If you can provide the complete problem statement for Problem 51, I'd be happy to provide a more specific solution!

Wrapping Up

So there you have it, guys! We've tackled some tough GP problems today, and hopefully, you've gained a better understanding of how to approach these types of questions. Remember, the key to mastering geometric progressions is practice, practice, practice! Keep working on problems, and you'll become a GP pro in no time. Keep practicing and exploring, and you'll be amazed at what you can achieve! Happy problem-solving!