Find First Four Terms Of Sequence An=46+8(n-1) Guide

Hey there, math enthusiasts! Ever stumbled upon a sequence and wondered how to crack it open? Well, you're in the right place. Today, we're diving deep into the world of sequences, specifically how to pinpoint the first four terms of a sequence defined by a formula. We'll break it down, step by step, making it super easy to follow along. So, grab your thinking caps, and let's get started!

The Sequence Unveiled: $a_n=46+8(n-1)$

Alright, let's kick things off by introducing the star of our show: the sequence $a_n=46+8(n-1)$. This might look a bit intimidating at first, but trust me, it's simpler than it seems. This formula is a blueprint, a recipe if you will, for generating the terms of our sequence. The $a_n$ part simply means "the nth term" – the term in the nth position. The 'n' is our variable, and it's going to take on values like 1, 2, 3, and so on. The rest of the formula, $46+8(n-1)$, is where the magic happens. It tells us exactly how to calculate each term based on its position in the sequence.

To truly grasp this, let's think about what this formula represents in plain English. It's saying, "To find any term in this sequence, start with 46, then add 8 times one less than the term's position." See? Not so scary after all! The key here is to understand that 'n' is just a placeholder. We'll plug in different values for 'n' to reveal the terms of our sequence, one by one. This process of substituting values into a formula is fundamental in mathematics, and it's something you'll encounter time and time again. So, mastering it now will set you up for success in all your future mathematical adventures. Remember, math isn't about memorizing formulas; it's about understanding the logic behind them. And in this case, the logic is beautifully simple: each term is built upon the previous one, with a consistent pattern of adding 8 (after the initial term). This kind of pattern is the essence of an arithmetic sequence, and it's what makes them so predictable and fascinating to study. So, with our formula in hand and our understanding solidified, let's move on to the exciting part: actually calculating the first four terms!

Decoding the First Term: $a_1$

Now, let's roll up our sleeves and find the first four terms, starting with the first term, denoted as $a_1$. Remember, the subscript '1' tells us we're looking for the term in the first position. To find $a_1$, we're going to substitute $n=1$ into our formula. This means we replace every 'n' in the formula with the number 1. So, $a_1$ becomes $46 + 8(1-1)$.

Let's break this down step by step, following the order of operations (PEMDAS/BODMAS). First, we tackle the parentheses: $(1-1)$ equals 0. Now our expression looks like this: $46 + 8(0)$. Next up is multiplication: $8(0)$ equals 0. So, we're left with $46 + 0$, which, of course, equals 46. Therefore, the first term of our sequence, $a_1$, is 46. And just like that, we've cracked the code for the first term! This seemingly simple calculation lays the foundation for understanding the entire sequence. It's the starting point, the anchor from which all other terms will be derived. Understanding how to find the first term is crucial because it often sets the pattern for the rest of the sequence. In this case, we know that 46 is our base, and the subsequent terms will build upon this foundation. This process of substitution and simplification is a core skill in algebra, and it's essential for solving a wide range of mathematical problems. So, pat yourself on the back for mastering this fundamental step! We've successfully navigated the formula, performed the necessary calculations, and arrived at our first term. With this momentum, let's move on to the next term and continue unraveling the secrets of this sequence.

Unveiling the Second Term: $a_2$

Great job on finding the first term! Now, let's keep the ball rolling and uncover the second term, $a_2$. Just like before, we'll use our trusty formula, but this time, we're substituting $n=2$. So, $a_2$ becomes $46 + 8(2-1)$. Ready to crunch the numbers?

Following the order of operations, we start with the parentheses: $(2-1)$ equals 1. This simplifies our expression to $46 + 8(1)$. Next, we take care of the multiplication: $8(1)$ equals 8. Now we're left with $46 + 8$, which adds up to 54. So, the second term, $a_2$, is 54. Fantastic! We're building our sequence term by term, and the pattern is starting to emerge. Notice how the second term is larger than the first? This hints at an increasing sequence, where each term gets progressively bigger. This kind of observation is a valuable skill in mathematics – the ability to spot patterns and make predictions based on initial calculations. It's like being a detective, piecing together clues to solve a mystery. In this case, the mystery is the nature of our sequence, and each term we find is a new piece of the puzzle. The fact that we're adding 8 in our formula directly translates to an increase of 8 between consecutive terms. This is a hallmark of arithmetic sequences, where the difference between terms remains constant. So, as we move on to calculate the next terms, we can use this knowledge to anticipate the results and double-check our work. Let's keep this momentum going and unveil the third term!

Decoding the Third Term: $a_3$

Alright, we're on a roll! We've successfully found the first two terms, and now it's time to tackle the third term, $a_3$. By now, you're probably getting the hang of this. We follow the same procedure: substitute $n=3$ into our formula. This gives us $a_3 = 46 + 8(3-1)$. Let's break it down!

First, the parentheses: $(3-1)$ equals 2. Our expression now reads $46 + 8(2)$. Next up, multiplication: $8(2)$ equals 16. This leaves us with $46 + 16$. Adding those together, we get 62. So, the third term, $a_3$, is 62. Woohoo! We're three terms down, and one to go. The sequence is steadily unfolding before our eyes. As we've calculated these terms, did you notice anything interesting? Each term is consistently 8 more than the previous one. This constant difference is a defining characteristic of an arithmetic sequence, and it's what makes them so predictable and easy to work with. In fact, once you know the first term and the common difference (in this case, 8), you can generate the entire sequence! This understanding not only helps us solve problems but also gives us a deeper appreciation for the elegance and structure of mathematics. So, with this insight in mind, let's move on to the final term in our quest, the fourth term, and complete our mission of unveiling the first four members of this fascinating sequence.

Revealing the Fourth Term: $a_4$

We've reached the final leg of our journey! Let's find the fourth term, $a_4$, and complete our mission. You know the drill: we substitute $n=4$ into our formula. This gives us $a_4 = 46 + 8(4-1)$. Let's get to work!

Starting with the parentheses: $(4-1)$ equals 3. Our expression becomes $46 + 8(3)$. Next, the multiplication: $8(3)$ equals 24. So, we have $46 + 24$, which adds up to 70. Therefore, the fourth term, $a_4$, is 70. Boom! We did it! We've successfully calculated all four terms. Pat yourselves on the back – you've officially become sequence sleuths! Now that we have all the pieces, let's take a step back and look at the bigger picture. We started with a formula, and through a series of simple substitutions and calculations, we've revealed the hidden pattern within. This process of taking a general rule and applying it to specific cases is a fundamental skill in mathematics and many other fields. It's how we move from abstract concepts to concrete results, from equations to real-world applications. And in this case, our "real-world" result is a sequence of numbers that tells a story of consistent growth and predictable progression. The journey of finding these four terms has not only given us the answers but also deepened our understanding of sequences and the power of mathematical formulas. So, with our mission accomplished, let's recap our findings and celebrate our success!

The First Four Terms: A Grand Finale

Drumroll, please! After all our hard work, it's time to unveil the first four terms of our sequence: 46, 54, 62, and 70. There you have it! We've successfully decoded the sequence and revealed its initial pattern. But more than just finding the numbers, we've also learned a valuable process. We've seen how a formula can act as a blueprint, guiding us step by step to uncover the terms of a sequence. We've practiced the art of substitution, plugging in different values to unlock different results. And we've reinforced the importance of the order of operations, ensuring we arrive at the correct answers. These skills are not just applicable to sequences; they're fundamental tools in the toolbox of any aspiring mathematician. And now, you possess them! So, take a moment to appreciate what you've accomplished. You've not only solved a problem but also deepened your understanding of mathematical concepts and techniques. And remember, the journey of learning math is not just about finding the right answers; it's about developing the skills and the confidence to tackle any challenge that comes your way. So, keep exploring, keep questioning, and keep pushing the boundaries of your mathematical knowledge. The world of numbers is vast and fascinating, and there's always something new to discover. And who knows, maybe the next sequence you encounter will be even more exciting than this one! Until then, keep practicing, keep learning, and most importantly, keep enjoying the beauty and power of mathematics.

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