Find U5, U6, U7, U20 In Sequence 3, 9, 18, 30...

Hey guys! Ever stumbled upon a sequence of numbers and felt like you're trying to crack a secret code? Well, you're not alone! Sequences are fascinating mathematical puzzles, and today we're going to dive deep into one: 3, 9, 18, 30... Our mission? To find the elusive U5, U6, U7, and even U20! So, buckle up, grab your thinking caps, and let's embark on this mathematical adventure together. We'll break down this sequence step-by-step, uncovering the pattern and ultimately finding those missing terms. This isn't just about crunching numbers; it's about understanding how these sequences work and building your problem-solving skills. Whether you're a student tackling homework or simply a curious mind, this exploration will equip you with the knowledge to conquer similar sequence challenges. So, let's get started and unlock the secrets hidden within this numerical pattern! Remember, math can be fun, especially when we approach it with curiosity and a willingness to learn. Let's transform this sequence from a mystery into a masterpiece of understanding. Think of sequences as numerical stories waiting to be told, and we're the storytellers ready to unravel their narratives. Each number holds a clue, and our job is to piece those clues together to reveal the bigger picture. By the end of this journey, you'll not only be able to find the specific terms we're looking for but also develop a keen eye for recognizing and working with various types of sequences. So, let's roll up our sleeves and get ready to decode the sequence 3, 9, 18, 30... together! This is going to be an exciting mathematical exploration, and I'm thrilled to have you along for the ride.

Cracking the Code: Identifying the Pattern

First things first, let's analyze the sequence: 3, 9, 18, 30.... To find U5, U6, U7, and U20, we need to figure out the pattern. What's happening between these numbers? Is it addition, subtraction, multiplication, division, or something more complex? Let's look at the differences between consecutive terms. The difference between 9 and 3 is 6. The difference between 18 and 9 is 9. The difference between 30 and 18 is 12. Hmm, the differences themselves (6, 9, 12) form a sequence! This suggests that the original sequence is not a simple arithmetic sequence (where the difference between terms is constant). Instead, it looks like the differences are increasing by a constant amount. This type of sequence is called a quadratic sequence. So, what does this mean for us? It means that the general formula for this sequence will involve a squared term (n^2). We're getting closer to cracking the code! This is the exciting part of mathematical problem-solving – piecing together the clues and forming a hypothesis. We've observed that the differences aren't constant, but the differences between the differences are constant (3 in this case). This is a key indicator of a quadratic sequence, and it gives us a significant head start in finding the general formula. Remember, the beauty of math lies in its patterns, and our ability to recognize these patterns is what empowers us to solve even the most challenging problems. So, with our newfound understanding of the sequence being quadratic, let's move on to the next step: finding the precise formula that governs this sequence. We're like mathematical detectives, uncovering the secrets hidden within these numbers!

Unveiling the Formula: Finding the General Term (Un)

Now that we suspect a quadratic sequence, let's find the general term (Un). A quadratic sequence has the general form: Un = an^2 + bn + c, where a, b, and c are constants we need to determine. To find these constants, we can use the first few terms of our sequence (3, 9, 18, 30). Let's substitute n = 1, 2, 3, and 4 into the general formula and create a system of equations. For n = 1, U1 = 3: a(1)^2 + b(1) + c = 3 => a + b + c = 3. For n = 2, U2 = 9: a(2)^2 + b(2) + c = 9 => 4a + 2b + c = 9. For n = 3, U3 = 18: a(3)^2 + b(3) + c = 18 => 9a + 3b + c = 18. We now have three equations with three unknowns (a, b, and c). We can solve this system of equations using various methods, such as substitution or elimination. Let's use elimination. Subtract the first equation from the second equation: (4a + 2b + c) - (a + b + c) = 9 - 3 => 3a + b = 6. Subtract the second equation from the third equation: (9a + 3b + c) - (4a + 2b + c) = 18 - 9 => 5a + b = 9. Now we have two equations with two unknowns: 3a + b = 6 and 5a + b = 9. Subtract the first of these equations from the second: (5a + b) - (3a + b) = 9 - 6 => 2a = 3 => a = 3/2. Substitute a = 3/2 back into 3a + b = 6: 3(3/2) + b = 6 => 9/2 + b = 6 => b = 6 - 9/2 => b = 3/2. Finally, substitute a = 3/2 and b = 3/2 back into a + b + c = 3: (3/2) + (3/2) + c = 3 => 3 + c = 3 => c = 0. So, we have a = 3/2, b = 3/2, and c = 0. Therefore, the general term for the sequence is Un = (3/2)n^2 + (3/2)n. We've cracked the code! This formula allows us to find any term in the sequence, no matter how far down the line. This is the power of finding the general term – it's like having a mathematical key that unlocks all the terms of the sequence. Finding the general term is a crucial step in understanding sequences, as it provides a concise and powerful way to represent the pattern. With this formula in hand, we're now ready to tackle the original challenge: finding U5, U6, U7, and U20. Let's move on to the next stage and put our formula to work!

The Grand Finale: Calculating U5, U6, U7, and U20

Now for the exciting part – using our formula Un = (3/2)n^2 + (3/2)n to find U5, U6, U7, and U20. Let's start with U5. Substitute n = 5 into the formula: U5 = (3/2)(5)^2 + (3/2)(5) = (3/2)(25) + (15/2) = 75/2 + 15/2 = 90/2 = 45. So, U5 = 45. Next, let's find U6. Substitute n = 6 into the formula: U6 = (3/2)(6)^2 + (3/2)(6) = (3/2)(36) + (18/2) = 108/2 + 18/2 = 126/2 = 63. So, U6 = 63. Now, let's calculate U7. Substitute n = 7 into the formula: U7 = (3/2)(7)^2 + (3/2)(7) = (3/2)(49) + (21/2) = 147/2 + 21/2 = 168/2 = 84. So, U7 = 84. Finally, let's find U20. Substitute n = 20 into the formula: U20 = (3/2)(20)^2 + (3/2)(20) = (3/2)(400) + (60/2) = 1200/2 + 60/2 = 1260/2 = 630. So, U20 = 630. There you have it! We've successfully found U5, U6, U7, and U20. This is the culmination of our mathematical journey – we started with a sequence, identified the pattern, found the general term, and finally, calculated the specific terms we were looking for. This process demonstrates the power of mathematical thinking and problem-solving. We've not only found the answers but also gained a deeper understanding of sequences and how they work. Remember, math is not just about memorizing formulas; it's about understanding the underlying concepts and applying them to solve problems. We've transformed this sequence from a puzzle into a solution, and that's something to celebrate! This is the essence of mathematical exploration – taking a challenge, breaking it down into manageable steps, and ultimately arriving at the answer. So, congratulations on cracking the code of this sequence! You've demonstrated the power of mathematical reasoning and problem-solving skills.

Conclusion: The Power of Patterns

So, guys, we've successfully navigated the sequence 3, 9, 18, 30... and found U5 = 45, U6 = 63, U7 = 84, and U20 = 630. But more importantly, we've learned how to approach sequence problems, identify patterns, and derive general formulas. This is a valuable skill that extends far beyond this specific example. The ability to recognize patterns and formulate mathematical models is crucial in various fields, from science and engineering to finance and computer science. Sequences are fundamental building blocks in mathematics, and understanding them opens doors to more advanced concepts. Whether you're working with arithmetic sequences, geometric sequences, or more complex quadratic sequences, the principles we've discussed here will serve you well. Remember, the key is to break down the problem, look for differences, identify the type of sequence, and then find the general term. Once you have the general term, you can find any term in the sequence with ease. Math is a journey of discovery, and each problem solved is a step forward in your mathematical understanding. So, keep exploring, keep questioning, and keep solving! The world of mathematics is full of fascinating patterns and challenges waiting to be uncovered. And with the skills you've developed here, you're well-equipped to tackle them. This is just the beginning of your mathematical adventure, and I encourage you to continue exploring the beauty and power of numbers. So, embrace the challenge, sharpen your problem-solving skills, and continue to unlock the secrets of the mathematical world! Remember, every problem is an opportunity to learn and grow, and the more you practice, the more confident and capable you'll become. So, go forth and conquer the world of sequences – the possibilities are endless!