Finding The Tenth Term In An Arithmetic Sequence

Hey everyone! Today, we're diving into the fascinating world of arithmetic sequences. Ever wondered how to predict the next number in a pattern? Well, arithmetic sequences are your answer! They're sequences where the difference between consecutive terms remains constant. It's like climbing a staircase where each step is the same height. So, grab your thinking caps, and let's unravel the mystery of finding the tenth term in a specific sequence.

Delving into Arithmetic Sequences

At its core, an arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. This constant difference is the unsung hero of our sequence, often called the common difference, which we affectionately denote as 'd'. Think of it as the rhythm section of our numerical orchestra, setting the pace for the entire sequence. In simpler terms, you start with a number, and then you keep adding the same value to get the next number, and the next, and so on. For example, the sequence 2, 5, 8, 11,... is an arithmetic sequence because we add 3 each time (our common difference!). Another way to imagine this, guys, is like a train chugging along a straight track, adding the same length of rail with each 'chug.' The 'chug' is the common difference in this analogy, keeping our train moving smoothly along the numerical tracks.

To put it into mathematical language, if we have the first term (which we call a_1) and the common difference (d), we can write out the entire sequence. The second term (a_2) is simply a_1 + d, the third term (a_3) is a_2 + d (which is the same as a_1 + 2d), and so on. See the pattern? We're just adding the common difference a certain number of times. This leads us to a brilliant shortcut: a formula that lets us jump straight to any term in the sequence without having to list out all the terms in between.

This formula is the golden key to unlocking any term in an arithmetic sequence. It looks like this:

a_n = a_1 + (n - 1)d

Where:

• a_n is the nth term we want to find
• a_1 is the first term of the sequence
• n is the position of the term we want (like 10th, 20th, or 100th)
• d is the common difference

This formula is like a mathematical GPS, guiding us directly to our desired term. It's built on the simple idea that to get to the nth term, we start with the first term and then add the common difference (n - 1) times. The '(n - 1)' part is crucial because we don't add the common difference to the first term itself; we add it to get to the next terms. So, to get to the 10th term, we add the common difference 9 times. Let's break down this formula with a real-world example. Imagine you're saving money. You start with $10 (a_1 = 10), and you save $5 every week (d = 5). How much money will you have after 12 weeks (n = 12)? Using our formula, a_12 = 10 + (12 - 1) * 5 = 10 + 55 = $65. See how powerful this formula is? It allows us to calculate values far down the sequence without having to tediously add the common difference over and over again. Isn't math amazing, guys?

Cracking the Code: Finding the Tenth Term

Now, let's tackle the original problem head-on. We're given that the first two terms of an arithmetic sequence are a_1 = 2 and a_2 = 4. Our mission, should we choose to accept it (and we do!), is to find a_10, the tenth term. It's like being given the first few clues in a puzzle and needing to find the final piece. Fear not, for we have the tools and the knowledge to solve this numerical mystery!

The first step, and often the most crucial, is to identify the common difference, 'd'. Remember, the common difference is the constant value added to each term to get the next. In our case, we can find 'd' by subtracting the first term from the second term. So, d = a_2 - a_1 = 4 - 2 = 2. This means that each term in our sequence is 2 more than the previous one. It's like climbing a ladder where each rung is exactly 2 inches higher than the last. Simple, right? But this simple step is the cornerstone of solving the entire problem. Now that we know 'd', we're one step closer to unlocking the tenth term.

Now that we've determined the common difference (d = 2), we can confidently use our golden formula to find the tenth term (a_10). Let's recap the formula: a_n = a_1 + (n - 1)d. We know a_1 is 2, n is 10 (because we want the tenth term), and d is 2. Plugging these values into the formula, we get:

a_10 = 2 + (10 - 1) * 2

Time for some good ol' arithmetic! First, we simplify the expression inside the parentheses: (10 - 1) = 9. Now we have:

a_10 = 2 + 9 * 2

Next, we perform the multiplication: 9 * 2 = 18. So our equation becomes:

a_10 = 2 + 18

Finally, we add the numbers together: 2 + 18 = 20. Voila! We've found it! The tenth term of the arithmetic sequence is 20. It's like reaching the top of a staircase and seeing the breathtaking view. Each step we took (identifying the common difference, applying the formula, and doing the arithmetic) led us to this satisfying conclusion. This whole process shows the power of breaking down a problem into smaller, manageable steps. By understanding the underlying principles (like the concept of common difference and the arithmetic sequence formula), we can tackle even seemingly complex problems with confidence.

The Answer and Its Significance

So, the answer to our question is a_10 = 20. We've successfully navigated the world of arithmetic sequences and pinpointed the tenth term. But what does this number really mean? It's more than just a solution; it's a glimpse into the continuing pattern of the sequence. Imagine listing out all the terms: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20... See how the sequence progresses? Each term faithfully follows the rule of adding 2, our common difference.

The fact that a_10 is 20 tells us that if we were to continue this pattern to the tenth position, we would land on the number 20. It's like knowing the final destination on a map. We know where we're going, and we know the path (adding 2 each time) to get there. This predictability is a key feature of arithmetic sequences and makes them incredibly useful in various real-world applications. Think about situations where things increase or decrease at a constant rate, like the depreciation of a car's value over time or the growth of a plant each day. Arithmetic sequences can help us model and understand these phenomena.

Moreover, understanding how to find specific terms in a sequence empowers us to make predictions. If we needed to know the 100th term, we wouldn't have to painstakingly list out 100 numbers. We could simply plug n = 100 into our formula and calculate it directly. This is the true power of mathematical tools; they allow us to leap over tedious calculations and gain insights into larger patterns. So, the next time you encounter a sequence of numbers, remember the magic of arithmetic sequences and the power of that little formula! It just might help you unlock some hidden patterns and make some valuable predictions. And that's something to celebrate, guys!

The final answer is (A) 20