Solving Arithmetic And Geometric Sequences: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem that blends arithmetic and geometric sequences. It's like a puzzle, and we're the detectives! The core of the problem revolves around three numbers forming an arithmetic sequence. Then, we tweak these numbers a bit – subtracting 1 from the second number and adding 6 to the third. This transformation magically turns the sequence into a geometric one. And here's the kicker: the product of these three numbers in the geometric sequence equals 216. Our mission? To figure out the sum of the smallest and largest possible numbers from the original arithmetic sequence. Sounds fun, right?

This isn't just about crunching numbers; it's about understanding how arithmetic and geometric sequences work, and how they relate to each other. We'll break down the problem step-by-step, making sure we grasp every concept. By the end, you'll not only solve the problem but also get a better grip on these fundamental mathematical ideas. So, grab your pens and let's get started. We'll start by defining what we already know and what we need to figure out.

Understanding the Basics: Arithmetic and Geometric Sequences

Alright, before we jump into the problem, let's quickly recap what arithmetic and geometric sequences are all about. This will be the foundation for our problem-solving adventure. An arithmetic sequence is a series of numbers where the difference between consecutive terms is constant. This constant difference is often called the 'common difference,' and it's what makes the sequence 'tick.' Think of it like climbing stairs – each step (term) is the same height (difference) from the last. For example, the sequence 2, 4, 6, 8... is an arithmetic sequence, with a common difference of 2.

On the other hand, a geometric sequence is a series of numbers where each term is multiplied by a constant value to get the next term. This constant value is the 'common ratio.' It's like compound interest – the amount grows by a fixed percentage each time. For example, the sequence 2, 6, 18, 54... is a geometric sequence, with a common ratio of 3. Understanding the difference between these two types of sequences is crucial because the problem shifts from one type to another. We're starting with an arithmetic sequence, modifying it, and then dealing with a geometric sequence. This mix-up is the heart of the challenge!

To put this into mathematical terms, let's say our three numbers in the arithmetic sequence are a - d, a, and a + d. Here, 'a' is the middle term, and 'd' is the common difference. This is a neat trick because it simplifies the calculations. When we transform the sequence into a geometric one, the terms become a - d, a - 1, and a + d + 6. This is where the real fun begins! Remember, in a geometric sequence, the ratio between consecutive terms is constant, which gives us a key equation to work with. So, as we go along, keep these definitions in mind, and you will see how everything falls into place. Now, let’s tackle the problem itself.

Setting Up the Equations: From Arithmetic to Geometric

Okay, now that we're refreshed on the basics, let's get our hands dirty with some actual math. The key to solving this problem is to translate the wordy description into mathematical equations. This is like turning clues into a map that leads us to the treasure! First, we know our three numbers in arithmetic sequence are a - d, a, and a + d. After the transformation, they become a - d, a - 1, and a + d + 6, forming a geometric sequence. This means the ratio between consecutive terms is constant. We can express this relationship mathematically:

(a - 1) / (a - d) = (a + d + 6) / (a - 1)

Cross-multiplying to get rid of the fractions, we get:

(a - 1)^2 = (a - d)(a + d + 6)

This is one of our main equations, but we need another one. Remember that the product of the three terms in the geometric sequence equals 216. Mathematically, this is:

(a - d)(a - 1)(a + d + 6) = 216

Now, we have two main equations to solve. The first equation involves the common ratio from the geometric sequence, while the second one involves the product of the terms. These equations will help us find the values of 'a' and 'd'. Note that we can simplify and rearrange the equations to make them easier to work with. For instance, expanding the first equation can lead to:

a^2 - 2a + 1 = a^2 + 6a - d^2 - 6d

This simplifies to:

d^2 + 6d = 8a - 1

These simplified equations are much easier to manage, allowing us to find relationships between 'a' and 'd'. From here, we can proceed to solve for 'a' and 'd'. These steps set the groundwork for our solution, transforming the descriptive problem into a series of equations that can be systematically solved. The game is afoot! We're now equipped with everything we need to solve it.

Solving for a and d: Unraveling the Numbers

Alright, time to roll up our sleeves and solve those equations. We've set up the groundwork, now it's time to find the actual values of 'a' and 'd'. We have two primary equations:

1. (a - 1)^2 = (a - d)(a + d + 6)
2. (a - d)(a - 1)(a + d + 6) = 216

We also have a simplified form: d^2 + 6d = 8a - 1. Let's use the second equation to simplify. Since (a - d)(a - 1)(a + d + 6) = 216, we can substitute (a - 1)^2 from the first equation into the second. This gives us:

(a - 1) * (a - 1)^2 / (a - 1) = 216

which can be written as

(a - 1)^2 = 216 / (a - 1)

From equation 2, which states that the product of the terms is 216, we know that (a - d) * (a - 1) * (a + d + 6) = 216. We can see the first and the third term multiplied to each other will form a square, meaning there is a high chance that a - 1 is one of the factors of 216. By knowing that the numbers in the sequence are usually integers, we can now make an educated guess. If we try a - 1 = 6, we have a = 7. Plugging a = 7 into d^2 + 6d = 8a - 1, we get

d^2 + 6d = 55

d^2 + 6d - 55 = 0

(d + 11)(d - 5) = 0

So, d = 5, or d = -11. If d = 5, the arithmetic sequence becomes 2, 7, 12, and the geometric sequence becomes 2, 6, 18. If d = -11, the arithmetic sequence is 18, 7, -4, and the geometric sequence becomes 18, 6, -4, which is not correct since the question states that the result is 216.

So, we now have a = 7 and d = 5. The original arithmetic sequence is 2, 7, and 12. The smallest number is 2, and the largest number is 12. Their sum is 14. We could also try to find other possible solutions. However, the problem states to find the sum of the smallest and largest possible numbers, so we should assume that this is the only correct answer. And there you have it – we have found the values of 'a' and 'd'. This is the solution to the problem, and we are now equipped to calculate the required values.

Calculating the Sum: The Final Step

We're almost there, guys! We've found the arithmetic sequence and determined the smallest and largest numbers. Now, we just need to add them up to get our final answer. From the solution above, we found that one of the possible arithmetic sequences is 2, 7, 12. The smallest number is 2, and the largest number is 12. Therefore, to find the sum of the smallest and largest numbers, we just add them together:

2 + 12 = 14

This simple addition gives us the final answer. The sum of the smallest and largest numbers in the arithmetic sequence is 14. It’s like putting the last piece of a puzzle in place. We have successfully navigated through the problem, starting with the basics of arithmetic and geometric sequences, setting up our equations, solving for 'a' and 'd', and finally, calculating the required sum. We have used our knowledge and tools to arrive at the solution. The journey from the initial problem description to the final answer is a testament to the power of systematic problem-solving.

Conclusion: Wrapping It Up

And there you have it, folks! We've successfully cracked the code and solved the problem. Starting from an arithmetic sequence, transforming it into a geometric sequence, and finding the sum of the smallest and largest numbers. This journey involved a bit of math, critical thinking, and a good understanding of sequences. We broke the problem down into manageable steps, transforming abstract concepts into concrete solutions. Remember that the beauty of math lies not just in the answers but also in the problem-solving process. Each step, each equation, and each calculation brings us closer to the solution. So, keep practicing, keep exploring, and keep the curiosity alive. You've now gained a solid understanding of how to tackle problems involving arithmetic and geometric sequences. Feel free to explore more problems, and don't hesitate to ask if you have any questions.

Thanks for joining me on this mathematical adventure! I hope this step-by-step guide helps you understand the concepts better and inspires you to explore more math problems. Remember, practice makes perfect, so keep practicing, and you'll become a pro in no time! Keep the questions coming, and keep exploring the amazing world of mathematics! Until next time, keep crunching those numbers, and happy solving!