Calculating The Sum Of An Arithmetic Progression First 15 Terms Example

Have you ever wondered how to calculate the sum of a series of numbers that follow a specific pattern? Well, today, we're diving into the fascinating world of arithmetic progressions! Specifically, we'll tackle the problem of finding the sum of the first 15 terms of an arithmetic progression where the first term is 50 and the common difference is -3. So, buckle up, math enthusiasts, let's get started!

Understanding Arithmetic Progressions

Before we jump into the calculation, let's quickly recap what an arithmetic progression actually is. 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. For example, the sequence 2, 4, 6, 8, 10 is an arithmetic progression with a common difference of 2. Each term is obtained by adding the common difference to the previous term. Think of it like climbing stairs, where each step is the same height (the common difference).

In our case, we have an arithmetic progression where the first term (often denoted as a₁) is 50 and the common difference (often denoted as d) is -3. This means the sequence starts at 50, and each subsequent term is 3 less than the previous one. So, the sequence would look something like this: 50, 47, 44, 41, and so on. Now, the question is, how do we find the sum of the first 15 terms of this sequence without having to manually add them all up? That's where the magic of formulas comes in!

The Formula for the Sum of an Arithmetic Progression

Fortunately, there's a neat formula that allows us to calculate the sum of the first n terms of an arithmetic progression. This formula is a real timesaver, especially when dealing with a large number of terms. The formula is:

Sₙ = (n/2) * [2a₁ + (n - 1)d]

Where:

• Sₙ is the sum of the first n terms
• n is the number of terms
• a₁ is the first term
• d is the common difference

This formula might look a little intimidating at first, but don't worry, it's actually quite straightforward. It essentially says that the sum of the first n terms is equal to half the number of terms, multiplied by the sum of twice the first term and the product of (n - 1) and the common difference. Let's break it down further with our specific problem.

Applying the Formula to Our Problem

Now, let's put this formula to work and calculate the sum of the first 15 terms of our arithmetic progression. Remember, we have:

• n = 15 (we want the sum of the first 15 terms)
• a₁ = 50 (the first term is 50)
• d = -3 (the common difference is -3)

Plugging these values into the formula, we get:

S₁₅ = (15/2) * [2(50) + (15 - 1)(-3)]

Let's simplify this step by step:

1. First, we calculate the term inside the brackets: 2(50) = 100 and (15 - 1)(-3) = 14(-3) = -42.
2. Now, we substitute these values back into the equation: S₁₅ = (15/2) * [100 + (-42)]
3. Simplify further: S₁₅ = (15/2) * [58]
4. Now, we multiply: S₁₅ = (15/2) * 58 = 15 * 29
5. Finally, we get the sum: S₁₅ = 435

So, the sum of the first 15 terms of the arithmetic progression is 435. Fantastic! We've successfully used the formula to solve the problem. You see, it's not as scary as it looks once you break it down. This formula is a powerful tool in your mathematical arsenal, allowing you to quickly calculate the sum of any arithmetic progression. It saves us from tedious manual calculations and gives us a clear path to the solution.

Why This Formula Works: A Glimpse Behind the Scenes

While we've successfully used the formula, you might be curious about why it works. Understanding the logic behind the formula can make it even more memorable and useful. The formula is based on the idea of pairing terms in the arithmetic progression. Imagine writing the series twice, once in the original order and once in reverse:

50 + 47 + 44 + ... + (the 15th term) (the 15th term) + ... + 44 + 47 + 50

If you add the corresponding terms vertically, you'll notice that each pair sums to the same value. This value is equal to the sum of the first term (a₁) and the last term (a₁₅). Since we have n terms, we have n/2 such pairs. Therefore, the sum of the entire series is (n/2) * (a₁ + a₁₅).

Now, to get our original formula, we need to express the last term (a₁₅) in terms of the first term and the common difference. The nth term of an arithmetic progression can be calculated as aₙ = a₁ + (n - 1)d. So, a₁₅ = 50 + (15 - 1)(-3) = 50 - 42 = 8. Substituting this back into our sum equation, we get:

S₁₅ = (15/2) * (50 + 8) = (15/2) * 58, which is exactly what we calculated earlier. This pairing method beautifully illustrates why the formula works and gives you a deeper understanding of the underlying mathematics.

Beyond the Formula: Real-World Applications of Arithmetic Progressions

Arithmetic progressions aren't just abstract mathematical concepts; they appear in various real-world scenarios. Recognizing these progressions can help you solve practical problems and make informed decisions. Let's explore some examples:

• Simple Interest: When you deposit money in a bank account that earns simple interest, the amount of interest earned each year forms an arithmetic progression. The first term is the initial interest earned, and the common difference is the constant interest amount added each year.
• Stacking Objects: Imagine stacking cans in a grocery store display. If each row has a fixed number of cans less than the row below it, the number of cans in each row forms an arithmetic progression. This concept can be used to optimize stacking arrangements.
• Salary Increments: In some jobs, employees receive a fixed salary increase each year. This annual increment creates an arithmetic progression, where the first term is the starting salary and the common difference is the increment amount. Understanding this progression can help employees plan their finances.
• Depreciation: The value of some assets, like cars, depreciates over time in a predictable way. If the depreciation amount is constant each year, the asset's value forms an arithmetic progression. This is important for accounting and financial planning.
• Patterns in Nature: Arithmetic progressions can even be found in natural phenomena. For example, the number of petals in some flowers can follow an arithmetic progression. While not all natural patterns are perfectly arithmetic, recognizing these patterns can give insights into the underlying processes.

By understanding arithmetic progressions, you can analyze and predict patterns in various situations, from finance to physics. It's a fundamental concept that empowers you to see the mathematical structure in the world around you. So, keep your eyes peeled – you never know where you might encounter an arithmetic progression next!

Practice Makes Perfect: Further Exploration

Now that you've learned how to calculate the sum of an arithmetic progression, the best way to solidify your understanding is to practice! Try solving similar problems with different values for the first term, common difference, and number of terms. You can even create your own arithmetic progressions and challenge yourself to find their sums. The more you practice, the more confident you'll become in using the formula and applying it to real-world scenarios. Math, like any skill, improves with consistent effort and practice.

Consider exploring other related concepts, such as geometric progressions (where terms are multiplied by a common ratio instead of adding a common difference) and series (the sum of the terms in a sequence). These concepts build upon the foundation of arithmetic progressions and open up new avenues for mathematical exploration. The world of mathematics is vast and interconnected, and each concept you learn strengthens your ability to understand and solve complex problems. So, keep learning, keep exploring, and keep having fun with numbers!

Conclusion: Mastering Arithmetic Progressions

So, there you have it! We've successfully calculated the sum of the first 15 terms of an arithmetic progression with a first term of 50 and a common difference of -3. We've explored the formula, understood its logic, and even discovered real-world applications of arithmetic progressions. Hopefully, this journey has demystified these sequences and shown you how powerful and practical they can be. Remember, math isn't just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively.

Keep practicing, keep exploring, and you'll be amazed at the mathematical insights you'll gain. Whether you're calculating interest on a loan, planning a stack of grocery cans, or simply appreciating the patterns in nature, arithmetic progressions and other mathematical concepts provide valuable tools for understanding and interacting with the world around us. So, embrace the challenge, enjoy the process, and keep those mathematical muscles flexed! You've got this!