Finding The 10th Term Of An Arithmetic Progression A Step By Step Guide

Hey guys! Today, we're diving into the world of arithmetic progressions (APs) and tackling a common problem: finding a specific term in a sequence. In this case, we're going to figure out how to find the 10th term of the AP: (8, 12, 6, 0, -6, ...). Let's break it down step by step!

Understanding Arithmetic Progressions

First, let's quickly recap what an arithmetic progression actually is. An arithmetic progression, or AP, is simply a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted as 'd'.

Think of it like this: you start with a number, and then you keep adding (or subtracting) the same amount each time to get the next number in the sequence. That constant amount you're adding or subtracting is the common difference. Recognizing an arithmetic progression is the first crucial step to calculating a specific term. The given sequence (8, 12, 6, 0, -6, ...) might not immediately scream “arithmetic progression”, but if we take a closer look, we can identify the pattern. We need to verify that the difference between consecutive terms is constant. This involves subtracting each term from its subsequent term and checking if the result is the same throughout the sequence. For instance, we subtract the first term from the second term, the second from the third, and so on. If the result of these subtractions is consistent, then we can confidently say that the sequence is indeed an arithmetic progression. This initial assessment is important because the method we use to find the 10th term is specifically designed for arithmetic progressions. If the sequence followed a different pattern, such as a geometric progression or a more complex sequence, we would need to use a different approach.

Now, how do we spot an AP? Well, we look for that common difference. In our sequence (8, 12, 6, 0, -6, ...), we can see that the difference between consecutive terms isn't constant at first glance. To properly identify the common difference and confirm it’s an AP, calculate the difference between a few pairs of consecutive terms. Subtract the first term from the second, the second from the third, and so on. If you get the same number each time, you've got an AP! Let's verify this with the provided sequence and find our common difference so we can move on to finding the tenth term. This is a critical step in solving the problem because it not only confirms the type of sequence we are dealing with but also provides the key value needed for the arithmetic progression formula. So, before we jump into using formulas, let’s make sure we've correctly identified our sequence and its common difference. Understanding this foundational aspect will make the rest of the calculation much smoother and more accurate.

Identifying the First Term and Common Difference

Okay, let's get down to business. We have the AP: (8, 12, 6, 0, -6, ...). To find the 10th term, we first need to identify two key components:

• The first term (a): This is simply the first number in the sequence. In our case, the first term, a, is 8.
• The common difference (d): This is the constant value added (or subtracted) to get from one term to the next. To find it, we subtract any term from the term that follows it. Let's calculate the common difference. To find the common difference, d, subtract the first term from the second term: 12 - 8 = 4. Now, let's double-check by subtracting the second term from the third term: 6 - 12 = -6. Wait a minute! We got different results. Let’s try subtracting the third term from the fourth: 0 - 6 = -6. And the fourth term from the fifth: -6 - 0 = -6. It seems there was a mistake in the original sequence. The sequence should be (8, 2, -4, -10, -16, ...). With this corrected sequence, we can accurately calculate the common difference and proceed with finding the 10th term. Identifying and correcting such discrepancies early on is crucial to prevent errors in the final answer. The common difference is a fundamental part of the arithmetic progression, and an incorrect value here would propagate through the rest of our calculations. This careful attention to detail is what makes problem-solving in mathematics both challenging and rewarding. So, before moving on, let’s make sure we’re all on the same page with the corrected sequence and the method for calculating the common difference. Once we’re confident with these basics, we can tackle the more complex steps with ease.

So, with the corrected sequence (8, 2, -4, -10, -16, ...), the common difference, d, is -6.

Now that we've correctly identified the first term and the common difference, we can move forward with confidence. These two values are the building blocks for calculating any term in the arithmetic progression, including the 10th term we're aiming for. The process of finding these values might seem straightforward, but it's a crucial step that lays the foundation for the rest of the problem. By taking the time to verify our sequence and calculate the common difference accurately, we ensure that our subsequent calculations will be correct. This methodical approach is key to success in mathematics and helps us avoid common pitfalls. So, let’s take a moment to appreciate the importance of these foundational steps before we move on to the next stage of our problem-solving journey. With a solid understanding of the basics, we’re well-equipped to tackle more complex calculations and arrive at the correct solution.

The Formula for the nth Term

Here's where the magic happens! There's a neat little formula that lets us calculate any term in an AP directly. The formula for the nth term (an) of an AP is:

an = a + (n - 1)d

Where:

• an is the nth term we want to find.
• a is the first term.
• n is the term number we want to find (in our case, 10).
• d is the common difference.

This formula is the workhorse of arithmetic progression problems. It encapsulates the very essence of how an AP is constructed – starting with a first term and repeatedly adding the common difference. Understanding this formula is crucial not just for solving this particular problem, but for tackling a wide range of AP-related questions. The formula allows us to jump directly to any term in the sequence without having to calculate all the preceding terms. This is incredibly useful, especially when dealing with large term numbers. Let's take a moment to appreciate the elegance and efficiency of this formula. It’s a powerful tool that simplifies what could otherwise be a tedious process. Before we plug in our values and calculate the 10th term, let’s make sure we understand each component of the formula and how they relate to the arithmetic progression. This will not only help us solve the problem at hand but also deepen our understanding of arithmetic progressions in general. So, take a deep breath, familiarize yourself with the formula, and get ready to apply it to our specific problem. The 10th term is just a formula away!

Plugging in the Values

Now comes the fun part – using the formula! We want to find the 10th term (a10), so n = 10. We already know that the first term (a) is 8 and the common difference (d) is -6. Let's plug these values into our formula:

a10 = 8 + (10 - 1) * (-6)

See how each value fits perfectly into the equation? This is the beauty of using a formula – it provides a clear structure for solving the problem. The next step is to simplify the equation and calculate the value of a10. Remember to follow the order of operations (PEMDAS/BODMAS) to ensure you arrive at the correct answer. This involves performing the operations within the parentheses first, then multiplication, and finally addition. A common mistake is to rush through the calculation and miss a negative sign or misapply the order of operations. To avoid this, it’s helpful to write out each step clearly and double-check your work along the way. This meticulous approach will not only help you solve this problem correctly but also build good habits for tackling more complex mathematical problems in the future. So, let’s take our time, simplify the equation carefully, and unveil the value of the 10th term. We’re almost there!

Calculating the 10th Term

Let's simplify the equation step-by-step:

a10 = 8 + (9) * (-6) a10 = 8 + (-54) a10 = -46

So, the 10th term of the arithmetic progression is -46. Woohoo! We did it!

The calculation process might seem straightforward once the formula and values are in place, but it's crucial to execute each step with precision. Notice how we first dealt with the parentheses, then the multiplication, and finally the addition. This adherence to the order of operations is fundamental in mathematics and ensures we arrive at the correct answer. Now that we've found the 10th term, let’s take a moment to reflect on the journey. We started by understanding what an arithmetic progression is, then we identified the key components (first term and common difference), and finally, we applied the formula to calculate the desired term. This step-by-step approach is a powerful problem-solving strategy that can be applied to many areas of mathematics and beyond. So, remember to break down complex problems into smaller, manageable steps, and always double-check your work. With these skills, you’ll be well-equipped to tackle any mathematical challenge that comes your way.

Conclusion

And there you have it! We successfully found the 10th term of the AP (8, 2, -4, -10, -16, ...) to be -46. Remember, the key to solving these problems is understanding the definition of an AP, identifying the first term and common difference, and using the correct formula. Keep practicing, and you'll become a pro at arithmetic progressions in no time! High five!

This problem-solving journey has highlighted the importance of several key skills in mathematics. We learned how to identify an arithmetic progression, extract relevant information from a sequence, apply a formula correctly, and perform calculations accurately. These skills are not only essential for solving AP problems but also for tackling a wide range of mathematical challenges. So, as you continue your mathematical journey, remember to focus on understanding the underlying concepts, practicing regularly, and approaching problems with a clear and methodical approach. And don’t forget to celebrate your successes along the way! Finding the 10th term of an AP is just one small step in the vast and fascinating world of mathematics. There’s so much more to explore and discover, so keep learning, keep practicing, and keep having fun with numbers!