Arithmetic Progression: Terms, Sums & Solutions

Hey math enthusiasts! Today, we're diving deep into the fascinating world of arithmetic progressions (APs). We'll be solving a classic problem where we're given some terms and asked to find the common difference, specific terms, and even the sum of terms. So, buckle up, grab your pens and papers, and let's get started! This exploration is designed to be super friendly and easy to follow, making it perfect for anyone looking to master APs. We will break down each step so that you can understand the process and easily solve problems like these in the future. We'll be using clear explanations, helpful examples, and a step-by-step approach to make sure you grasp every concept. This is a must-know topic for anyone studying mathematics, and we're here to make sure you have a solid understanding of all the key concepts. Whether you're a student preparing for an exam or just someone curious about math, this guide will provide you with the knowledge and confidence to tackle AP problems with ease. Let's make learning math a fun and rewarding experience! Remember, practice is key, so don't hesitate to work through the examples and try some additional problems on your own. You'll be surprised at how quickly you can master arithmetic progressions when you approach them with curiosity and determination. Keep in mind that understanding the fundamentals is crucial. Once you have a firm grasp of the basic concepts, you'll be able to solve complex problems with confidence. So, let's unlock the secrets of arithmetic progressions together! This article aims to equip you with the knowledge and skills needed to excel in your mathematical journey. Let's explore the intricacies of sequences and series, uncovering their hidden patterns and applications.

Finding the Common Difference in an Arithmetic Progression

Alright, let's kick things off by figuring out the common difference in our arithmetic progression. The problem tells us that the fourth term (a₄) is -3 and the ninth term (a₉) is 12. Remember, in an arithmetic progression, the difference between consecutive terms is constant, and this constant difference is what we call the common difference, often denoted by 'd'. To find the common difference, we can use the following formula which is derived from the core definition of APs: aₙ = a₁ + (n - 1)d. We know the value of two terms, so we can use these two terms to find the common difference. We can write the fourth term (a₄) as a₁ + 3d = -3 and the ninth term (a₉) as a₁ + 8d = 12. To solve for 'd', we can subtract the equation for the fourth term from the equation for the ninth term. So, (a₁ + 8d) - (a₁ + 3d) = 12 - (-3). This simplifies to 5d = 15. Now, just divide both sides by 5 to get the common difference, d = 3. So, the common difference of this AP is 3. This means that each term in the sequence increases by 3. Understanding how to find the common difference is fundamental. It lays the groundwork for solving other related problems, such as finding specific terms or the sum of a series. Mastering this skill will not only help you solve the given problem, but it will also strengthen your overall understanding of arithmetic progressions. Feel confident as you start solving more complex problems. Remember, the key is to apply the right formulas and techniques. Also, don't be afraid to break down the problem into smaller, more manageable steps. This will make the entire process less daunting and much easier to follow. With each problem you solve, your understanding and confidence in arithmetic progressions will continue to grow. Embrace the journey and enjoy the process of learning.

Unveiling the Fifth Term of the Sequence

Now, let's find the fifth term (a₅) of this AP. Since we now know the common difference (d = 3) and the fourth term (a₄ = -3), it's a piece of cake! To get the fifth term, we simply add the common difference to the fourth term: a₅ = a₄ + d. Substituting the values we have, a₅ = -3 + 3. So, the fifth term, a₅, is 0. Easy peasy, right? The beauty of arithmetic progressions is that once you understand the pattern and have key information like the common difference, finding any term becomes straightforward. The formula aₙ = a₁ + (n - 1)d is your best friend here. But in this case, we could directly add the common difference to the fourth term since we needed to find the consecutive term. Always remember the relationship between terms in an AP. They follow a consistent pattern. Understanding this will allow you to quickly and accurately determine any term in the sequence. By understanding the relationship between consecutive terms, you can greatly simplify your calculations. Always think about how different terms relate to one another. This mental approach makes solving problems much more efficient. Keep in mind that a good understanding of fundamental concepts can empower you to tackle even the most challenging problems with confidence. Remember, the goal is to not only find the right answer but also to understand the underlying principles.

Determining the Number of Terms for a Specific Sum

Finally, we want to find the number of terms (n) that will give a sum of 135. We know the sum of an arithmetic series (Sₙ) can be calculated using the formula: Sₙ = (n/2) * [2a₁ + (n - 1)d]. Here, Sₙ = 135, d = 3, and we need to find 'n'. First, we need to find the first term (a₁) to use the formula. We know a₄ = -3 and d = 3, so using a₄ = a₁ + 3d, we get -3 = a₁ + 3(3), which simplifies to a₁ = -12. Now, let's plug the known values into the sum formula: 135 = (n/2) * [2(-12) + (n - 1)3]. Simplifying, we get 135 = (n/2) * [-24 + 3n - 3]. This becomes 135 = (n/2) * [3n - 27]. Multiplying both sides by 2, we get 270 = n(3n - 27), and this expands to 270 = 3n² - 27n. Now, divide everything by 3: 90 = n² - 9n. Rearranging to form a quadratic equation, we have n² - 9n - 90 = 0. Factoring this quadratic equation, we get (n - 15)(n + 6) = 0. So, n = 15 or n = -6. Since the number of terms cannot be negative, we discard -6. Therefore, the number of terms that give a sum of 135 is n = 15. The ability to work with the sum of an arithmetic series opens up another layer of understanding. Be sure to carefully evaluate the given information. Then, apply the relevant formulas, and solve for the unknown variables. The sum of the series is a critical concept to understand. It enables you to find the total value of the terms in the series. Remember, when you solve an equation, it's always good practice to double-check your answer to avoid any errors. In this case, we found the number of terms. This means we have successfully solved all the parts of the question. You've now gained all the necessary skills to calculate various aspects of an arithmetic progression. Congratulations! You've successfully navigated through finding the common difference, specific terms, and the number of terms needed to achieve a specific sum in an arithmetic progression. Keep practicing to solidify your skills!