Finding The 10th Term Of A Geometric Sequence A(n)=-3(2)^(n-1)

Hey guys! Ever stumbled upon a geometric sequence and felt a bit lost trying to find a specific term? Don't worry, we've all been there. Geometric sequences might sound intimidating, but they're actually pretty straightforward once you grasp the basics. In this article, we're going to break down the process of finding the 10th term of the geometric sequence a(n) = -3(2)^(n-1). We'll not only solve this particular problem but also equip you with the knowledge to tackle any similar sequence question that comes your way. So, let's dive in and make geometric sequences your new best friend!

Understanding Geometric Sequences

First things first, let's define what a geometric sequence actually is. In simple terms, a geometric sequence is a list of numbers where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio. Think of it as a chain reaction, where each number is linked to the one before it by this ratio. For example, the sequence 2, 4, 8, 16... is a geometric sequence because each term is obtained by multiplying the previous term by 2 (the common ratio). Understanding this fundamental concept is crucial before we proceed to tackle the problem at hand. Why? Because the entire formula we'll be using hinges on this idea of a common ratio and its consistent application throughout the sequence.

Now, let's dig a bit deeper into the key components of a geometric sequence. The first term, often denoted as a₁, is the starting point of the sequence. It's the seed from which all other terms grow. The common ratio, usually represented by r, is the multiplier that dictates how the sequence progresses. It's the engine driving the sequence forward. With these two elements, a₁ and r, we can define any geometric sequence. The beauty of a geometric sequence lies in its predictable nature. Once you identify the first term and the common ratio, you can, in theory, find any term in the sequence, no matter how far down the line it is. This predictability is what makes geometric sequences so useful in various mathematical and real-world applications. From calculating compound interest to modeling population growth, the principles of geometric sequences are surprisingly versatile. So, as we move forward, keep in mind the importance of identifying a₁ and r – they are the keys to unlocking the secrets of any geometric sequence.

To solidify your understanding, let's consider a few more examples. The sequence 5, 10, 20, 40... is geometric with a₁ = 5 and r = 2. The sequence 1, -3, 9, -27... is also geometric, but with a₁ = 1 and r = -3. Notice that the common ratio can be negative, which results in alternating signs in the sequence. This adds another layer of nuance to geometric sequences, making them even more interesting to explore. Now, let's think about how this knowledge applies to the problem we're trying to solve. We have the geometric sequence defined by a(n) = -3(2)^(n-1). Can you identify the first term and the common ratio in this case? Take a moment to ponder this question. Recognizing these elements in the given formula is the first step towards finding the 10th term. In the next section, we'll delve into the general formula for finding the nth term of a geometric sequence, which will provide us with the tools we need to solve this problem systematically.

The General Formula for Geometric Sequences

Alright, now that we've got a solid grip on what geometric sequences are, let's arm ourselves with the ultimate weapon for finding any term in the sequence: the general formula. This formula is the cornerstone of working with geometric sequences, and it's going to be our best friend in solving the problem at hand. The general formula for the nth term (aₙ) of a geometric sequence is given by:

aₙ = a₁ * r^(n-1)

Where:

• aₙ is the nth term (the term we want to find)
• a₁ is the first term of the sequence
• r is the common ratio
• n is the term number (the position of the term in the sequence)

This formula might look a bit intimidating at first glance, but trust me, it's incredibly powerful and easy to use once you break it down. Let's dissect each component of the formula to understand its role. a₁, as we discussed earlier, is the starting point of the sequence. It's the foundation upon which the entire sequence is built. The common ratio, r, is the multiplier that determines how the sequence progresses. It's the factor that connects each term to the next. The exponent, (n-1), is crucial because it reflects the fact that the first term (a₁) doesn't need to be multiplied by the common ratio. It's already the starting point. For the second term, we multiply by r once; for the third term, we multiply by r twice, and so on. This is why we have (n-1) in the exponent.

Now, let's see how this formula works in practice. Suppose we have a geometric sequence with a₁ = 3 and r = 2. If we want to find the 5th term (a₅), we simply plug in the values into the formula: a₅ = 3 * 2^(5-1) = 3 * 2⁴ = 3 * 16 = 48. See? It's as easy as plugging in the values and doing the math. This formula allows us to bypass the need to calculate each term individually. Instead of finding the 2nd term, then the 3rd, then the 4th, and finally the 5th, we can jump directly to the 5th term using the formula. This is particularly useful when dealing with large term numbers, like finding the 10th or 20th term. Imagine having to manually calculate each term up to the 20th – that would be incredibly tedious! The general formula saves us time and effort, allowing us to focus on the core concepts. But remember, the power of the formula lies in understanding the underlying principles of geometric sequences. Without a solid grasp of a₁, r, and n, the formula becomes just a meaningless jumble of symbols. So, make sure you understand the meaning behind each element before you start plugging in values. With this understanding, you'll be well-equipped to tackle any geometric sequence problem that comes your way. In the next section, we'll apply this formula to the specific problem we're trying to solve: finding the 10th term of the sequence a(n) = -3(2)^(n-1).

Before we move on, let's reinforce our understanding with another example. Consider the sequence where a₁ = 10 and r = 0.5. If we want to find the 8th term, we'll use the formula aₙ = a₁ * r^(n-1). Plugging in the values, we get a₈ = 10 * (0.5)^(8-1) = 10 * (0.5)⁷ = 10 * 0.0078125 = 0.078125. This example demonstrates that the common ratio doesn't have to be a whole number; it can be a fraction or a decimal as well. The formula works just the same, regardless of the nature of r. The key is to identify the correct values of a₁, r, and n, and then apply the formula meticulously. Now, with these examples under our belt, we're more than ready to tackle the original problem. We've dissected the general formula, understood its components, and seen how it works in practice. In the next section, we'll put all of this knowledge to use and finally find the 10th term of the geometric sequence a(n) = -3(2)^(n-1).

Solving for the 10th Term

Okay, folks, the moment we've been preparing for! Let's finally put our knowledge to the test and find the 10th term of the geometric sequence a(n) = -3(2)^(n-1). We've got the general formula in our arsenal, and we've honed our understanding of geometric sequences. Now it's time to execute. The first step, as always, is to identify the key components: a₁, r, and n. Looking at the given formula, a(n) = -3(2)^(n-1), we can easily extract these values.

• a₁: This is the first term, which is the coefficient multiplying the exponential part. In this case, a₁ = -3.
• r: This is the common ratio, which is the base of the exponential term. Here, r = 2.
• n: We're looking for the 10th term, so n = 10.

Now that we have these values, it's simply a matter of plugging them into the general formula: aₙ = a₁ * r^(n-1). Substituting the values, we get:

a₁₀ = -3 * 2^(10-1)

Now, let's simplify this expression step by step. First, we deal with the exponent: 10 - 1 = 9. So, we have:

a₁₀ = -3 * 2⁹

Next, we calculate 2⁹. If you have a calculator handy, this is a breeze. If not, you can break it down: 2⁹ = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 = 512. So, we have:

a₁₀ = -3 * 512

Finally, we multiply -3 by 512:

a₁₀ = -1536

And there we have it! The 10th term of the geometric sequence a(n) = -3(2)^(n-1) is -1536. You did it! By systematically identifying the first term, common ratio, and term number, and then applying the general formula, we successfully found the 10th term. This process highlights the power of understanding the underlying principles and applying the formula correctly. It's not just about memorizing a formula; it's about understanding how it works and why it works. With this understanding, you can confidently tackle any geometric sequence problem.

Before we celebrate our victory, let's take a moment to reflect on the process. We started by defining geometric sequences and understanding the key concepts of first term and common ratio. Then, we introduced the general formula and dissected its components. Finally, we applied the formula to solve the problem at hand. This step-by-step approach is crucial for success in mathematics. It's about breaking down complex problems into smaller, manageable steps. So, remember this approach as you continue your mathematical journey. Now, let's move on to the final step: choosing the correct answer from the given options.

Identifying the Correct Answer

We've done the hard work, guys! We've calculated the 10th term of the geometric sequence a(n) = -3(2)^(n-1) and found it to be -1536. Now, the final step is to match our answer with the given options. Let's take a look at the options:

A. -54 B. -1,536 C. -3,072 D. -10,077,696

It's clear that our calculated answer, -1536, corresponds to option B. So, the correct answer is B. -1,536. This step might seem trivial, but it's essential to ensure that you're selecting the correct option. Sometimes, you might make a small calculation error or misread the options, so it's always a good idea to double-check your answer and make sure it matches one of the choices.

Now, let's think about why the other options are incorrect. Option A, -54, is significantly smaller in magnitude than our answer. This suggests that it might be the result of a misunderstanding of the exponential growth in a geometric sequence. Option C, -3,072, is double our answer. This could be a result of multiplying by the common ratio an extra time, or perhaps a calculation error in the exponent. Option D, -10,077,696, is a very large negative number. This indicates a significant error in the calculation, possibly involving raising 2 to a much higher power than 9. Analyzing why the incorrect options are wrong can be a valuable learning experience. It helps you identify potential pitfalls and avoid similar errors in the future. It's not just about getting the right answer; it's about understanding the process and recognizing common mistakes.

So, to recap, we successfully found the 10th term of the geometric sequence, matched it with the correct option, and analyzed why the other options were incorrect. This comprehensive approach demonstrates a thorough understanding of geometric sequences and problem-solving techniques. You guys are now well-equipped to tackle similar problems with confidence. Remember, the key is to understand the concepts, apply the formulas correctly, and double-check your work. With practice, you'll become a geometric sequence master!

In this article, we've taken a deep dive into the world of geometric sequences and successfully found the 10th term of the sequence a(n) = -3(2)^(n-1). We started by establishing a solid understanding of geometric sequences, identifying the key components of first term and common ratio. We then armed ourselves with the general formula for finding the nth term and dissected its components. By applying this formula systematically, we calculated the 10th term to be -1536 and correctly identified option B as the answer.

But more importantly, we've learned a valuable problem-solving approach that can be applied to various mathematical challenges. We've emphasized the importance of understanding the underlying concepts, applying formulas correctly, and analyzing potential errors. This holistic approach will not only help you excel in geometric sequences but also empower you to tackle any mathematical problem with confidence. So, keep practicing, keep exploring, and keep challenging yourselves. The world of mathematics is full of fascinating patterns and concepts waiting to be discovered. And remember, the journey of learning is just as important as the destination. Until next time, happy problem-solving, guys!