Calculating The Sum Of An Arithmetic Series: S = 3 + 6 + ... + 63

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Calculating the Sum of an Arithmetic Series: s = 3 + 6 + ... + 63

Hey guys! Today, we're diving into a classic math problem: calculating the sum of an arithmetic series. Specifically, we're tackling the series s = 3 + 6 + 9 + ... + 63. This might seem daunting at first, but don't worry! We'll break it down step by step, so you can easily understand how to solve it. Whether you're a student prepping for an exam or just a math enthusiast, you're in the right place. Let's get started and make math a little less mysterious together!

Understanding Arithmetic Series

First things first, let's understand what an arithmetic series actually is. An arithmetic series is simply the sum of an arithmetic sequence. Now, what's an arithmetic sequence? It’s a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference. In our series, 3 + 6 + 9 + ... + 63, the common difference is 3 because each term is 3 more than the previous term. Recognizing this pattern is crucial for solving the problem.

Why is understanding arithmetic series important? Well, these kinds of series pop up all over the place, from simple financial calculations (like figuring out how much you save each month if you increase your savings by a fixed amount) to more complex physics problems (like calculating the distance an object travels when it accelerates at a constant rate). So, mastering this concept is a valuable skill. Think of it as adding another tool to your problem-solving toolbox.

Now, let’s look at our specific series: 3 + 6 + 9 + ... + 63. We know the first term (a) is 3 and the common difference (d) is also 3. But how many terms are there in the series? This is a key piece of information we need to figure out before we can calculate the sum. To find the number of terms, we need to understand the relationship between the terms in the sequence. Each term can be expressed as a + (n-1)d, where 'n' is the term number. So, the last term, 63, can be expressed in this form. Let's use this to find 'n'.

Finding the Number of Terms

To find the number of terms in the series, we need to figure out how many times we add the common difference (3) to the first term (3) to reach the last term (63). We can use the formula for the nth term of an arithmetic sequence: a_n = a_1 + (n - 1)d, where:

  • a_n is the nth term (in our case, 63)
  • a_1 is the first term (in our case, 3)
  • n is the number of terms (what we want to find)
  • d is the common difference (in our case, 3)

Plugging in the values, we get: 63 = 3 + (n - 1)3. Now, let's solve for n:

  1. Subtract 3 from both sides: 60 = (n - 1)3
  2. Divide both sides by 3: 20 = n - 1
  3. Add 1 to both sides: 21 = n

So, there are 21 terms in the series. Now that we know this, we're one step closer to finding the sum. This is a crucial step because without knowing the number of terms, we can't use the formula for the sum of an arithmetic series. Think of it like trying to bake a cake without knowing how many eggs to use – it just won't turn out right! Now that we have n, we can confidently move on to the next part of the problem.

Knowing the number of terms is like having a complete map before starting a journey. It gives us a clear understanding of the scope of the problem and allows us to apply the correct formula. Without it, we'd be wandering in the dark. So, always remember to find the number of terms first when dealing with arithmetic series. It’s a fundamental step that makes the rest of the solution much easier to handle. Let's proceed to the next step where we'll finally calculate the sum of the series using a neat formula.

Calculating the Sum Using the Formula

Now for the fun part: calculating the sum! There's a nifty formula for the sum of an arithmetic series, which makes our lives much easier. The formula is: S_n = n/2 * (a_1 + a_n), where:

  • S_n is the sum of the first n terms
  • n is the number of terms
  • a_1 is the first term
  • a_n is the last term

We've already found all these pieces of the puzzle! We know n = 21, a_1 = 3, and a_n = 63. Let's plug these values into the formula:

  • S_21 = 21/2 * (3 + 63)
  • S_21 = 21/2 * 66
  • S_21 = 10.5 * 66
  • S_21 = 693

So, the sum of the series s = 3 + 6 + 9 + ... + 63 is 693. How cool is that? We’ve successfully navigated through the problem, step by step, and arrived at the answer. This formula is a powerful tool, guys. It allows us to quickly calculate the sum of any arithmetic series, no matter how long it is. It's like having a shortcut through a mathematical maze!

The formula works by essentially pairing the first and last terms, the second and second-to-last terms, and so on. Each of these pairs adds up to the same value (in our case, 3 + 63 = 6 + 60 = 9 + 57 = 66). The formula then multiplies this value by half the number of terms (n/2) to get the total sum. It’s a beautiful and efficient way to solve these kinds of problems. Remember this formula; it will serve you well in many mathematical adventures!

Alternative Method: Pairing Terms

While the formula is super efficient, let's explore another way to think about this problem. This method helps to build a deeper understanding of why the formula works. We can manually pair the terms in the series and add them up. This approach is particularly useful for smaller series, but it also gives us insight into the structure of arithmetic series in general. Think of it as understanding the engine instead of just knowing how to drive the car!

Here’s how we can do it:

  1. Pair the first and last terms: 3 + 63 = 66
  2. Pair the second and second-to-last terms: 6 + 60 = 66
  3. Pair the third and third-to-last terms: 9 + 57 = 66

Notice a pattern? Each pair sums up to 66. This is because as we move inwards from the ends of the series, one term increases by the common difference (3), and the other term decreases by the common difference (3), so their sum remains constant. Now, how many such pairs do we have?

Since we have 21 terms, we'll have 10 pairs that sum up to 66, and one middle term left over. The middle term is the 11th term, which is 3 + (11 - 1) * 3 = 3 + 30 = 33. So, the sum can be calculated as (10 * 66) + 33. Let's check it out:

  • (10 * 66) + 33 = 660 + 33 = 693

Voila! We arrived at the same answer using a different method. This approach not only confirms our previous result but also provides a more intuitive understanding of the series. It shows us that the sum of an arithmetic series can be thought of as the average of the first and last term, multiplied by the number of terms. This pairing method is a fantastic way to visualize and grasp the concept behind the formula. It’s like seeing the gears turning inside the clock, rather than just looking at the time!

Real-World Applications

You might be wondering,