Finding S_26 In An Arithmetic Progression Given A_12 + A_15 = 20

Hey guys! Today, we're diving into a fun problem involving arithmetic progressions. Arithmetic progressions are sequences where the difference between consecutive terms is constant. Think of it like a staircase where each step is the same height. Our mission? To find the sum of the first 26 terms (S_26) of an arithmetic progression, given that the 12th term (a_12) plus the 15th term (a_15) equals 20. Sounds like a puzzle, right? Let's break it down step by step and make it super clear. We'll use the properties of arithmetic progressions to link the given information to what we need to find. So, grab your thinking caps, and let's get started!

Understanding Arithmetic Progressions

Before we jump into solving the problem, let's quickly recap what arithmetic progressions are all about. An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'. The general form of an arithmetic progression is: a, a + d, a + 2d, a + 3d, ... where 'a' is the first term. So, each term is simply the previous term plus the common difference. This simple pattern is the key to solving many problems related to these sequences. For example, if our first term 'a' is 2 and the common difference 'd' is 3, the progression would look like: 2, 5, 8, 11, and so on. Each number is 3 more than the one before it. This constant addition makes arithmetic progressions predictable and easy to work with, which is why they pop up in various math problems and real-world scenarios. Whether it's calculating simple interest or modeling the depreciation of an asset, understanding arithmetic progressions gives you a powerful tool for solving a wide array of problems. So, let's keep this basic definition in mind as we tackle the challenge of finding S_26!

Key Formulas for Arithmetic Progressions

To effectively solve our problem, we need to be familiar with a couple of key formulas related to arithmetic progressions. These formulas will help us connect the terms and sums in the sequence. Firstly, the n-th term (a_n) of an arithmetic progression can be found using the formula: a_n = a + (n - 1)d where 'a' is the first term, 'd' is the common difference, and 'n' is the term number. This formula is super handy because it allows us to find any term in the sequence without having to list out all the terms before it. Just plug in the values, and bam, you've got your answer! Secondly, the sum of the first n terms (S_n) of an arithmetic progression can be calculated using the formula: S_n = n/2 * [2a + (n - 1)d] or alternatively, S_n = n/2 * [a + a_n] where 'a' is the first term, 'd' is the common difference, 'n' is the number of terms, and a_n is the n-th term. Both versions of the formula are useful, depending on what information you have available. The first version is great when you know 'a' and 'd', while the second one is perfect if you know 'a' and the last term a_n. Knowing these formulas is like having a secret weapon in your math arsenal. They make solving problems involving arithmetic progressions much more straightforward and less time-consuming. So, let's keep these formulas close as we move on to applying them to our specific problem!

Setting Up the Equations

Alright, guys, let's get down to the nitty-gritty of our problem. We're given that a_12 + a_15 = 20, and our goal is to find S_26. The first step in cracking this is to use the formula for the n-th term of an arithmetic progression to express a_12 and a_15 in terms of the first term 'a' and the common difference 'd'. Remember, the formula is a_n = a + (n - 1)d. So, for a_12, we have: a_12 = a + (12 - 1)d = a + 11d And for a_15, we have: a_15 = a + (15 - 1)d = a + 14d Now, we know that a_12 + a_15 = 20, so we can substitute our expressions for a_12 and a_15 into this equation: (a + 11d) + (a + 14d) = 20 This simplifies to: 2a + 25d = 20 This is our first equation, and it links the first term 'a' and the common difference 'd'. Now, to find S_26, we'll use the formula for the sum of the first n terms: S_n = n/2 * [2a + (n - 1)d] For S_26, this becomes: S_26 = 26/2 * [2a + (26 - 1)d] = 13 * [2a + 25d] Notice anything interesting? The expression inside the brackets, 2a + 25d, is exactly what we found in our first equation! This is a crucial connection that will help us solve the problem. By setting up these equations, we've laid the groundwork for finding S_26. We've translated the given information into mathematical expressions, and now we're ready to put the pieces together.

Solving for S_26

Okay, so we've got our equations set up, and now it's time to bring it all home and solve for S_26. Remember, we found that 2a + 25d = 20, and we want to find S_26, which we expressed as S_26 = 13 * [2a + 25d]. This is where the magic happens! We can directly substitute the value of 2a + 25d from our first equation into the formula for S_26. So, we have: S_26 = 13 * (20) This is a straightforward calculation: S_26 = 260 And there you have it! The sum of the first 26 terms of the arithmetic progression is 260. Isn't it cool how everything just clicked into place? By using the properties and formulas of arithmetic progressions, we were able to link the given information to what we needed to find. This is a classic example of how breaking down a problem into smaller steps and using the right tools can lead to a clear and elegant solution. So, next time you're faced with a similar challenge, remember to identify the key relationships, set up your equations, and take it one step at a time. You've got this!

Conclusion

Fantastic job, guys! We've successfully navigated through this arithmetic progression problem and found that S_26 = 260. We started by understanding the basic concepts of arithmetic progressions and their key formulas. Then, we translated the given information into mathematical equations, which allowed us to connect the 12th and 15th terms to the sum of the first 26 terms. Finally, by substituting and simplifying, we arrived at our answer. This problem beautifully illustrates how mathematical concepts can be applied to solve real problems. It’s not just about memorizing formulas; it’s about understanding how they relate to each other and using them strategically. Arithmetic progressions, like many mathematical topics, have a wide range of applications, from finance to physics. Mastering these concepts not only helps in exams but also equips you with valuable problem-solving skills that can be used in various fields. So, keep practicing, keep exploring, and keep challenging yourself with new problems. The more you engage with these concepts, the more confident and skilled you'll become. And remember, every problem solved is a step forward in your mathematical journey. Keep up the awesome work!