Finding The 12th Term In A Geometric Sequence Step-by-Step Solution

Hey there, math enthusiasts! Ever stumbled upon a geometric sequence that seemed like a puzzle? Well, let's dive into one together! We've got a geometric sequence where the 4th term ($a_4$) is 54 and the 7th term ($a_7$) is a whopping 1,458. The mission, should you choose to accept it, is to find the 12th term ($a_{12}$). Buckle up, because we're about to decode this sequence!

Understanding Geometric Sequences

Before we jump into solving, let's quickly recap what geometric sequences are all about. In a geometric sequence, each term is multiplied by a constant value to get the next term. This magic number is called the common ratio, often denoted as 'r'. Think of it like this: you start with a number, and then you multiply it by 'r', then multiply the result by 'r' again, and so on. The general formula for the nth term ($a_n$) of a geometric sequence is:

$a_n = a_1 * r^(n-1)$

Where $a_1$ is the first term and 'n' is the term number. Now that we've got the basics down, let's tackle our problem. In geometric sequences, the common ratio is the linchpin, and our first task is to unearth this value. Remember, the common ratio is the constant multiplier that bridges each term to its successor. With a bit of algebraic finesse, we can extract this crucial piece of information from the given terms. The beauty of geometric sequences lies in their inherent structure, where each term is a multiple of the previous one. This multiplicative relationship allows us to set up equations and solve for the unknowns, such as the common ratio. So, let's roll up our sleeves and delve into the process of finding 'r', the key to unlocking the mystery of this geometric sequence. By understanding how terms relate to each other through the common ratio, we gain a powerful tool for predicting and analyzing the sequence's behavior. This step is not just about crunching numbers; it's about grasping the underlying pattern that governs the sequence's progression. Once we have the common ratio in hand, we can then use it to navigate through the sequence, effortlessly calculating any term we desire. This is the essence of working with geometric sequences – mastering the common ratio opens up a world of possibilities for exploration and discovery. So, let's focus our energy on this fundamental step, as it lays the groundwork for our ultimate goal of finding the 12th term. It’s like finding the right key to unlock a treasure chest – once we have it, the rest will follow naturally.

Finding the Common Ratio (r)

We know $a_4 = 54$ and $a_7 = 1458$. Using the general formula, we can write these as:

$a_4 = a_1 * r^3 = 54$ $a_7 = a_1 * r^6 = 1458$

Now, let's divide the second equation by the first:

$(a_1 * r^6) / (a_1 * r^3) = 1458 / 54$

Notice how the $a_1$ terms cancel out, leaving us with:

$r^3 = 27$

Taking the cube root of both sides, we get:

$r = 3$

Awesome! We've found the common ratio. Now that we've pinpointed the common ratio, we're holding a powerful key that unlocks the secrets of this geometric sequence. It's like discovering the secret ingredient in a recipe – suddenly, everything starts to come together. With r = 3, we can now trace the sequence's path, understanding how each term springs from its predecessor. The elegance of geometric sequences lies in this consistent multiplicative relationship, where the common ratio acts as the bridge between terms. Now, the challenge is to harness this knowledge and apply it strategically to find the 12th term, which is our ultimate goal. Remember, the common ratio isn't just a number; it's the heartbeat of the sequence, dictating its rhythm and growth. By mastering its value, we gain a profound understanding of the sequence's behavior and can confidently predict its future terms. So, let's celebrate this milestone – finding the common ratio is a crucial step forward, and it sets the stage for the exciting calculations that lie ahead. It's like reaching a landmark on a treasure map, knowing that we're getting closer to the final destination. With r = 3 in our grasp, let's proceed with confidence and unravel the mystery of the 12th term. It's a journey of mathematical discovery, and we're well-equipped to conquer it!

Finding the First Term ($a_1$)

To find $a_{12}$, we need the first term, $a_1$. We can use either $a_4$ or $a_7$ and the common ratio we just found. Let's use $a_4$:

$54 = a_1 * 3^3$ $54 = a_1 * 27$ $a_1 = 54 / 27$ $a_1 = 2$

Great! We've nailed down the first term. Now that we've successfully determined the first term, $a_1$, we've added another vital piece to our geometric sequence puzzle. It's like finding the starting point on a map – now we have a clear origin from which to chart our course. The first term serves as the foundation upon which the entire sequence is built, and knowing its value allows us to accurately calculate any term down the line. Think of it as the seed from which the sequence grows, with the common ratio acting as the fertilizer that drives its expansion. With both $a_1$ and 'r' in our toolkit, we're now fully equipped to tackle the final challenge: finding the 12th term. It's a testament to the power of breaking down a problem into manageable steps – by systematically uncovering each component, we've arrived at a point where the solution is within easy reach. This is the essence of mathematical problem-solving: methodical exploration and strategic application of concepts. So, let's take a moment to appreciate this milestone – we've navigated the intricacies of the geometric sequence and emerged with a clear understanding of its underlying structure. Now, with confidence and precision, let's proceed to the final act: calculating the elusive 12th term. It's the culmination of our efforts, and the reward for our diligent pursuit of knowledge.

Calculating the 12th Term ($a_{12}$)

Now that we have $a_1 = 2$ and $r = 3$, we can find $a_{12}$ using the formula:

$a_{12} = a_1 * r^(12-1)$ $a_{12} = 2 * 3^11$ $a_{12} = 2 * 177147$ $a_{12} = 354294$

So, the 12th term of the geometric sequence is 354,294. Boom! We've successfully navigated the twists and turns of this geometric sequence and arrived at our destination: the 12th term. It's a moment of triumph, a testament to our perseverance and problem-solving skills. The 12th term, a hefty 354,294, stands as a testament to the exponential growth inherent in geometric sequences. It's a dramatic illustration of how a constant multiplier, the common ratio, can lead to rapid escalation in value. This journey has not only given us a numerical answer but also a deeper appreciation for the elegance and power of mathematical patterns. We've seen how the interplay between the first term and the common ratio shapes the entire sequence, dictating its trajectory and magnitude. Now, equipped with this newfound understanding, we can confidently tackle other geometric sequence challenges, unraveling their mysteries and predicting their behavior. It's a feeling of empowerment that comes from mastering a mathematical concept and applying it successfully. So, let's bask in the satisfaction of a job well done – we've conquered this problem with skill and precision, and we're ready to embrace the next mathematical adventure that comes our way. The world of numbers is vast and fascinating, and we've just taken another step in our journey of exploration.

Conclusion

Therefore, the 12th term of the geometric sequence is 354,294. Great job, guys! You've successfully navigated the world of geometric sequences and emerged victorious. Remember, the key to solving these problems lies in understanding the core concepts – the common ratio, the first term, and the general formula. With these tools in your arsenal, you're well-equipped to tackle any geometric sequence challenge that comes your way. So, keep exploring, keep learning, and keep pushing your mathematical boundaries. The world of numbers is full of fascinating patterns and mysteries, just waiting to be uncovered. And who knows? Maybe the next geometric sequence you encounter will lead you to even greater mathematical discoveries. So, stay curious, stay engaged, and never stop the quest for knowledge. The beauty of mathematics is that it's a never-ending journey of learning and exploration, and every problem you solve brings you one step closer to a deeper understanding of the universe around us. So, congratulations on this achievement, and may your mathematical adventures continue to be filled with excitement and success!