Arithmetic Sequence: Find The Next Term & Sum

Hey guys! Let's dive into the fascinating world of arithmetic sequences. This topic is super important in mathematics, and understanding it can open doors to solving some really cool problems. We're going to break down a specific sequence, figure out its next term, and then calculate the sum of its first 28 terms. Sounds like a plan? Let's get started!

Unveiling the Arithmetic Sequence: 6, 11, 16...

Identifying the Pattern

In the world of sequences, arithmetic sequences hold a special place. These sequences follow a predictable pattern, making them relatively easy to work with. The key characteristic of an arithmetic sequence is that the difference between consecutive terms remains constant. This constant difference is called the common difference. To figure out if the given sequence, 6, 11, 16..., is an arithmetic sequence, we need to check if the difference between successive terms is the same. Let's calculate the difference between the second term (11) and the first term (6), which is 11 - 6 = 5. Now, let's do the same for the third term (16) and the second term (11): 16 - 11 = 5. Bingo! The difference is the same in both cases, which means we've confirmed that this is indeed an arithmetic sequence with a common difference of 5. Understanding this fundamental concept is crucial because it allows us to predict future terms in the sequence and perform various calculations, such as finding the sum of a certain number of terms. Spotting this pattern is the first step in mastering arithmetic sequences.

(a) Finding the Next Term

Now that we know the sequence 6, 11, 16... is an arithmetic sequence with a common difference of 5, finding the next term becomes a piece of cake! Remember, in an arithmetic sequence, each term is obtained by adding the common difference to the previous term. So, to find the term that comes after 16, we simply add 5 to it: 16 + 5 = 21. Therefore, the next term in the sequence is 21. This simple addition showcases the beauty of arithmetic sequences – their predictable nature makes them easy to work with. You can continue this process to find subsequent terms as well. For example, the term after 21 would be 21 + 5 = 26, and so on. This predictability is what makes arithmetic sequences so useful in various mathematical applications. Identifying the common difference is the key to unlocking the sequence's pattern and predicting future terms. Keep practicing, and you'll become a pro at spotting these patterns!

(b) Calculating the Sum of the First 28 Terms

Okay, guys, this is where things get a little more interesting! We're tasked with finding the sum of the first 28 terms of our arithmetic sequence. Now, we could manually add up all 28 terms, but that would take ages and be prone to errors. Thankfully, there's a nifty formula specifically designed for calculating the sum of an arithmetic series (which is the sum of the terms in an arithmetic sequence). The formula is: Sn = n/2 [2a + (n - 1)d], where: Sn represents the sum of the first n terms, n is the number of terms we want to sum, a is the first term of the sequence, and d is the common difference. We already know a = 6 (the first term), d = 5 (the common difference), and we want to find the sum of the first 28 terms, so n = 28. Let's plug these values into the formula: S28 = 28/2 [2(6) + (28 - 1)5]. Now, it's just a matter of simplifying the expression. First, we have 28/2 = 14. Then, 2(6) = 12, and (28 - 1)5 = 27 * 5 = 135. So, our equation becomes: S28 = 14 [12 + 135]. Next, we add 12 and 135 to get 147. Finally, we multiply 14 by 147, which gives us S28 = 2058. Therefore, the sum of the first 28 terms of the arithmetic sequence is 2058. Isn't that formula a lifesaver? It allows us to quickly and accurately calculate the sum of arithmetic series without having to manually add up a large number of terms.

Breaking Down the Sum Formula for Arithmetic Series

Understanding the Formula's Components

The formula Sn = n/2 [2a + (n - 1)d] might look a bit intimidating at first, but let's break it down to understand what each part represents and why it works. This formula is the key to efficiently calculating the sum of an arithmetic series, so grasping its components is crucial. The first part, Sn, simply denotes the sum of the first 'n' terms of the arithmetic sequence. It's what we're trying to find! The 'n' itself represents the number of terms we want to add up. For instance, in our previous example, we wanted to find the sum of the first 28 terms, so 'n' was 28. Next, we have 'a', which stands for the first term of the sequence. In our example, the first term was 6. The 'd' represents the common difference between consecutive terms. As we established earlier, the common difference in our sequence is 5. Now, let's look at the expression inside the brackets: [2a + (n - 1)d]. The term '2a' represents twice the first term. The term '(n - 1)d' represents the difference between the first term and the nth term. Think of it as adding the common difference (d) to the first term (n-1) times to reach the nth term. Finally, the entire expression is multiplied by n/2. This is essentially averaging the first and last terms and then multiplying by the number of terms. This formula beautifully captures the essence of arithmetic series, allowing us to calculate the sum without tedious manual addition.

Why Does the Formula Work?

Now that we've identified the components of the formula, let's delve into why it works. The beauty of this formula lies in its ability to efficiently calculate the sum by pairing terms in a clever way. Imagine writing out the arithmetic series: a, a + d, a + 2d, ..., a + (n-2)d, a + (n-1)d. Now, let's write the same series in reverse: a + (n-1)d, a + (n-2)d, ..., a + d, a. If we add these two series together term by term, we'll notice something remarkable. The sum of the first terms (a + [a + (n-1)d]) is the same as the sum of the second terms ([a + d] + [a + (n-2)d]), and so on. Each pair of terms adds up to 2a + (n-1)d. Since we have 'n' terms in the series, we'll have n such pairs. Therefore, the sum of the two series is n[2a + (n-1)d]. But remember, we added the series to itself, so this sum is twice the sum of the original series (2Sn). To get the sum of the original series (Sn), we simply divide by 2: Sn = n/2 [2a + (n - 1)d]. This pairing strategy is a classic mathematical trick that allows us to derive a concise formula for the sum of an arithmetic series. By understanding the logic behind the formula, we gain a deeper appreciation for its elegance and efficiency.

Applying the Formula in Different Scenarios

The sum formula for arithmetic series is not just a theoretical concept; it's a powerful tool that can be applied in various real-world scenarios. Let's explore some examples to illustrate its versatility. Imagine you're planning to save money each month, increasing your savings by a fixed amount. For instance, you might save $100 in the first month, $110 in the second month, $120 in the third month, and so on. This forms an arithmetic sequence where the first term (a) is 100 and the common difference (d) is 10. Using the sum formula, you can easily calculate the total amount you'll save over a certain period, say, 12 months. Another example could involve calculating the total number of seats in a stadium where each row has a fixed number of seats more than the row in front of it. If the first row has 20 seats and each subsequent row has 2 more seats, you can use the formula to find the total seating capacity of the stadium. Furthermore, the formula can be used in physics to calculate the distance traveled by an object with constant acceleration. These examples highlight the practical applications of the arithmetic series sum formula in various fields, making it a valuable tool in problem-solving. Mastering this formula will empower you to tackle a wide range of mathematical challenges.

Wrapping Up: Mastering Arithmetic Sequences

So, guys, we've journeyed through the world of arithmetic sequences, from identifying their patterns to calculating the sum of their terms. We started with a specific sequence, 6, 11, 16..., and successfully found the next term and the sum of its first 28 terms. We also delved into the sum formula, understanding its components and why it works. Remember, the key to mastering arithmetic sequences is practice. Work through various examples, try to identify the common difference, and apply the sum formula in different scenarios. The more you practice, the more comfortable you'll become with these concepts. Arithmetic sequences are a fundamental building block in mathematics, and a solid understanding of them will pave the way for tackling more advanced topics. Keep exploring, keep practicing, and most importantly, keep having fun with math! You've got this!