Geometric Progressions Calculations Problems And Solutions
Hey guys! Let's dive into the fascinating world of Geometric Progressions (GPs). We're going to break down some common problems, making sure you not only understand the math but also feel confident tackling these questions on your own. So, buckle up, and let's get started!
Decoding Geometric Progressions
Before we jump into the calculations, let's quickly recap what a Geometric Progression actually is. In simple terms, a GP is a sequence of numbers where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio (often denoted as 'r'). Think of it like a snowball rolling down a hill – it gets bigger and bigger at an increasing rate. That's the power of a GP!
Key Concepts in Geometric Progressions
To really nail these problems, it's essential to understand the core concepts of geometric progressions. The most important concept to grasp is the common ratio. The common ratio, denoted as 'r', is the cornerstone of any GP. It's the constant factor by which each term is multiplied to obtain the next term. Mathematically, if you have a sequence like a, ar, ar², ar³, and so on, 'r' is what links them all together. This understanding is crucial because many GP problems hinge on finding or using this ratio.
Another fundamental concept is the general term of a GP, often written as an = a1 * r^(n-1). Here, 'an' is the nth term, 'a1' is the first term, 'r' is the common ratio, and 'n' is the term number. This formula is your go-to tool for finding any term in a GP without having to list out all the preceding terms. It’s especially useful when dealing with large term numbers, like the 10th or 15th term, as we’ll see in some of our examples.
Furthermore, recognizing a GP from a given sequence is key. You can quickly check if a sequence is a GP by verifying whether the ratio between consecutive terms is constant. For example, if you have a sequence 2, 6, 18, 54, you can divide 6 by 2 (which gives 3), 18 by 6 (also 3), and 54 by 18 (again, 3). Since the ratio is consistently 3, you know this is a GP. This simple check can save you a lot of time and prevent errors in more complex problems.
Knowing these basics like the back of your hand will not only help you solve problems faster but also give you a deeper appreciation for the elegance and patterns within mathematical sequences. So, let's keep these concepts in mind as we tackle our first set of problems. Remember, understanding these fundamentals is like having the right tools in your toolbox – it makes every job easier!
Problem 38 Finding 'x' in Geometric Progressions
Our first challenge involves finding the value of 'x' in different geometric progressions. This type of problem usually involves setting up equations based on the property that the ratio between consecutive terms in a GP is constant. Let's break down each part step by step.
Part a) (x+1, x+9, x+15)
In this sequence, (x+1, x+9, x+15), we need to ensure that the ratio between the first and second terms is equal to the ratio between the second and third terms. In other words, (x+9) / (x+1) should be equal to (x+15) / (x+9). This sets up our equation:
(x+9) / (x+1) = (x+15) / (x+9)
To solve this, we'll cross-multiply, which means multiplying (x+9) by (x+9) and (x+1) by (x+15). This gives us:
(x+9)² = (x+1)(x+15)
Expanding both sides, we get:
x² + 18x + 81 = x² + 16x + 15
Now, we simplify by subtracting x² from both sides and rearranging the terms:
18x + 81 = 16x + 15
Subtracting 16x from both sides gives:
2x + 81 = 15
Subtracting 81 from both sides yields:
2x = -66
Finally, dividing both sides by 2, we find:
x = -33
So, for this sequence to be a GP, x must be -33.
Part b) (x-3, x, x+6)
Next up, we have the sequence (x-3, x, x+6). We apply the same principle here: the ratio between consecutive terms must be constant. This means x / (x-3) should equal (x+6) / x. Setting up the equation:
x / (x-3) = (x+6) / x
Cross-multiplying gives us:
x² = (x-3)(x+6)
Expanding the right side:
x² = x² + 3x - 18
Subtracting x² from both sides:
0 = 3x - 18
Adding 18 to both sides:
18 = 3x
Dividing both sides by 3, we get:
x = 6
Therefore, for this sequence to be a GP, x must be 6.
Part c) (x, x+9, x+45)
Lastly, we tackle the sequence (x, x+9, x+45). Again, we set up the equation based on the common ratio: (x+9) / x should equal (x+45) / (x+9). The equation is:
(x+9) / x = (x+45) / (x+9)
Cross-multiplying gives:
(x+9)² = x(x+45)
Expanding both sides:
x² + 18x + 81 = x² + 45x
Subtracting x² from both sides:
18x + 81 = 45x
Subtracting 18x from both sides:
81 = 27x
Dividing both sides by 27, we find:
x = 3
Thus, for this sequence to form a GP, x must be 3.
By solving these equations, we've successfully found the values of 'x' that make each sequence a geometric progression. Remember, the key is to set up the ratios correctly and then use algebraic manipulation to solve for the unknown. You got this!
Problem 39 Calculating Terms in a Geometric Progression
Now, let's shift our focus to calculating specific terms in a geometric progression. This usually involves using the formula for the nth term of a GP, which we touched on earlier. This formula is a powerful tool for finding any term in the sequence without having to list out all the preceding ones.
Understanding the Formula
Before we dive into the calculations, let's quickly revisit the formula for the nth term of a GP: an = a1 * r^(n-1). Remember, 'an' is the nth term we want to find, 'a1' is the first term of the sequence, 'r' is the common ratio, and 'n' is the term number. This formula is your best friend when dealing with GP problems, so make sure you're comfortable with it.
To effectively use this formula, it's important to first identify the first term (a1) and the common ratio (r) from the given GP. The first term is usually straightforward – it's simply the first number in the sequence. The common ratio, as we discussed earlier, is found by dividing any term by its preceding term. Once you have these two values, plugging them into the formula along with the desired term number (n) is all you need to do.
But why does this formula work so well? It essentially captures the compounding nature of a GP. The common ratio 'r' is raised to the power of (n-1) because the first term doesn't need to be multiplied by 'r'; it's the starting point. Each subsequent term is then the result of multiplying by 'r' one additional time. Understanding this logic can help you remember the formula and apply it correctly in various scenarios.
Moreover, this formula isn’t just a tool for finding specific terms; it also provides insights into the growth pattern of the GP. A common ratio greater than 1 indicates exponential growth, meaning the terms get larger and larger. Conversely, a common ratio between 0 and 1 indicates exponential decay, where the terms get smaller and smaller. A negative common ratio means the terms alternate in sign, creating an oscillating pattern. Recognizing these patterns can give you a deeper understanding of the sequence and help you predict future terms more intuitively.
So, as we move forward to calculate the 10th, 15th, and nth terms of our given GP, remember that this formula is your key. Let's break down the given sequence and apply this formula to find the terms we need. By understanding not just the how but also the why behind this formula, you'll be well-equipped to tackle any GP term calculation!
The Sequence (1, 2, 4, 8, ...)
We're given the geometric progression (1, 2, 4, 8, ...). From this, we can easily identify the first term, a1, as 1. To find the common ratio, r, we divide any term by its preceding term. For instance, 2 / 1 = 2, 4 / 2 = 2, and 8 / 4 = 2. So, the common ratio, r, is 2.
Calculating the 10th Term
To find the 10th term, we use the formula an = a1 * r^(n-1) with n = 10. Plugging in the values, we get:
a10 = 1 * 2^(10-1) = 1 * 2^9 = 1 * 512 = 512
So, the 10th term of the GP is 512.
Calculating the 15th Term
Similarly, for the 15th term, we use n = 15:
a15 = 1 * 2^(15-1) = 1 * 2^14 = 1 * 16384 = 16384
Thus, the 15th term of the GP is 16,384.
Calculating the nth Term
Finally, to find the nth term, we simply leave 'n' as a variable in our formula:
an = 1 * 2^(n-1) = 2^(n-1)
This gives us the general expression for the nth term of the sequence. It’s a concise way to represent any term in the sequence based on its position, n.
By calculating these terms, we've shown the power of the general term formula in geometric progressions. Whether you're finding a specific term or a general expression, this formula is your go-to tool. Practice using it, and you'll become a GP pro in no time!
Problem 40 Expressing the General Term of a Succession
Now, let's tackle the challenge of writing the expression for the general term of a succession. This is a fundamental skill in dealing with sequences and series, as it allows us to define any term in the sequence based on its position. Essentially, we're looking for a formula that will give us any term if we plug in its term number.
Identifying Patterns
The first step in finding the general term is to carefully observe the sequence and identify any patterns. This might involve looking at the differences between consecutive terms, the ratios between terms, or any other relationships that stand out. For a geometric progression, as we've been discussing, the key pattern to look for is the common ratio.
When analyzing the sequence, ask yourself: Is it arithmetic, where the difference between consecutive terms is constant? Is it geometric, where the ratio between consecutive terms is constant? Or does it follow some other pattern? Sometimes, sequences can be a combination of different patterns, or they might follow a more complex rule. The more you practice, the better you'll become at spotting these patterns quickly.
Once you've identified a pattern, the next step is to express this pattern mathematically. This often involves finding a formula that relates the term number (n) to the value of the term. For GPs, we already know the general form of the formula: an = a1 * r^(n-1). But for other types of sequences, you might need to derive the formula from scratch. This could involve trial and error, algebraic manipulation, or even some educated guessing.
Moreover, understanding the different types of sequences and their general forms is incredibly helpful. Arithmetic sequences have a linear general term, geometric sequences have an exponential general term, and there are many other types of sequences with their own characteristic forms. Knowing these general forms can guide your search for the specific formula for a given sequence.
Remember, expressing the general term is like writing a recipe for the sequence. It’s a formula that tells you exactly what to do to get any term you want. So, let’s put these pattern-spotting skills to the test and find the general term for our sequence!
I cannot answer question 40 because the sequence is missing.
Final Thoughts on Geometric Progressions
We've covered a lot in this guide, from the basics of geometric progressions to solving specific problems involving finding values of 'x', calculating terms, and expressing general terms. Remember, the key to mastering GPs is understanding the core concepts, especially the common ratio and the general term formula.
Practice is crucial! The more problems you solve, the more comfortable you'll become with identifying patterns, setting up equations, and applying the formulas. Don't be afraid to make mistakes – they're part of the learning process. Each mistake is an opportunity to understand the concept better.
And remember, math isn't just about formulas and calculations; it's about understanding the underlying logic and patterns. The more you understand the 'why' behind the 'how', the more confident and skilled you'll become. So, keep exploring, keep practicing, and keep unlocking the mysteries of mathematics!
