Finding The Position Of 222 In The Arithmetic Sequence 282, 276, 270

Hey guys! Ever stumbled upon a sequence that just makes you scratch your head? Well, today, we're going to unravel one such mystery together. We've got this sequence: 282, 276, 270, and the big question is: where does the number 222 fit in? What position does it hold in this numerical lineup? Sounds like a fun puzzle, right? So, let's put on our math hats and get cracking!

Understanding the Arithmetic Progression

To figure out the position of 222, we first need to understand what kind of sequence we're dealing with. Looking at the numbers, we can see that the difference between consecutive terms is constant. Specifically, 276 - 282 = -6 and 270 - 276 = -6. This tells us we're working with an arithmetic progression (AP). An arithmetic progression, for those who might need a refresher, is a sequence where the difference between any two successive members is a constant. This constant difference is what we call the common difference, often denoted by 'd'. In our case, the common difference (d) is -6. The first term of the sequence, which we'll call 'a', is 282. Now that we know it's an arithmetic progression, we can use the formula for the nth term of an AP to help us find the position of 222. This formula is our secret weapon in solving this puzzle, so let's keep it handy. It's all about identifying patterns and using the right tools, and in math, formulas are definitely our best tools! Think of it like this: we've got a map (the formula) and a destination (the position of 222). All we need to do is follow the map!

The Formula for the Nth Term

The formula for the nth term (an) of an arithmetic progression is given by: an = a + (n - 1)d. Where: an is the nth term of the sequence, a is the first term, n is the position of the term in the sequence (what we're trying to find!), d is the common difference. This formula is the key to unlocking the mystery of where 222 sits in our sequence. It's like a decoder ring for numbers! We know a, we know d, and we know the value of the term we're interested in (222). The only thing missing is 'n', the position. So, we're going to plug in the values we know and solve for 'n'. It's like a mathematical treasure hunt, and 'n' is the treasure! Let's break it down step-by-step. We're not just blindly plugging in numbers; we're understanding how each part of the formula relates to the sequence. This is what makes math so powerful – it's not just about memorizing formulas, it's about understanding the logic behind them. When we truly understand the 'why', the 'how' becomes much easier.

Applying the Formula to Our Sequence

In our sequence, we have a = 282, d = -6, and we want to find the position (n) when an = 222. So, we can plug these values into the formula: 222 = 282 + (n - 1)(-6). Now, it's just a matter of solving for 'n'. This is where our algebra skills come into play. We're going to simplify the equation, isolate 'n', and voilà, we'll have our answer. Think of it like untangling a knot. We carefully work through each step, undoing the operations until we have 'n' all by itself. Remember, the key to algebra is to do the same thing to both sides of the equation to keep it balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other to keep it level. Let's walk through the steps together. First, we'll distribute the -6. Then, we'll combine like terms. Finally, we'll isolate 'n' by adding or subtracting and then dividing. And just like that, the mystery will be solved! It's like detective work, but with numbers instead of clues.

Solving for 'n'

Let's solve the equation step-by-step: 222 = 282 + (n - 1)(-6). First, distribute the -6: 222 = 282 - 6n + 6. Next, combine like terms: 222 = 288 - 6n. Now, isolate the term with 'n' by subtracting 288 from both sides: 222 - 288 = -6n, which simplifies to -66 = -6n. Finally, divide both sides by -6 to solve for 'n': n = -66 / -6, which gives us n = 11. So, the number 222 is in the 11th position in the sequence. Wasn't that satisfying? We took a seemingly complex problem and broke it down into manageable steps. This is the beauty of math – it gives us a systematic way to solve problems, no matter how daunting they may seem at first. We started with a sequence, identified its pattern, used a formula, and solved for the unknown. It's like a recipe – follow the steps, and you'll get the right result every time. And now, we know exactly where 222 fits in our sequence. High five!

Verification

To be absolutely sure, let's verify our answer. We can plug n = 11 back into the formula: a11 = 282 + (11 - 1)(-6) = 282 + (10)(-6) = 282 - 60 = 222. Bingo! It checks out. This step is super important because it's like double-checking your work on a test. It gives you the confidence that you've got the right answer. And in math, confidence is key! It's not enough to just get an answer; we want to be sure that it's the correct answer. That's why verification is such a valuable tool. It's like having a safety net – it catches you if you've made a mistake along the way. So, always remember to verify your solutions, guys. It's the mark of a true math pro!

Conclusion

So, there you have it! The number 222 occupies the 11th position in the sequence 282, 276, 270... We successfully navigated this numerical puzzle by understanding the concept of arithmetic progressions and wielding the mighty formula for the nth term. Remember, guys, math isn't about memorizing; it's about understanding. And by understanding the underlying principles, we can conquer any mathematical challenge that comes our way. This journey of finding the position of 222 wasn't just about getting the right answer; it was about the process of problem-solving. We learned how to identify patterns, apply formulas, and verify our results. These are skills that will serve us well not just in math class, but in life in general. So, keep exploring, keep questioning, and keep solving! And remember, math can be fun, especially when we tackle it together!