Calculating The Sum Of The Series 1 + 3 + 5 + ... + 99

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Calculating the Sum of the Series 1 + 3 + 5 + ... + 99

Hey guys! Today, we're diving into a fun math problem: calculating the sum of the series S = 1 + 3 + 5 + ... + 99. This might seem daunting at first, but don't worry, we'll break it down step-by-step and make it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Series

Before we jump into calculations, let's first understand what this series is all about. The series S = 1 + 3 + 5 + ... + 99 consists of the sum of all odd numbers from 1 up to 99. Recognizing this pattern is the first step to solving the problem efficiently. We need to figure out how many terms are in this series and then use a formula to calculate the sum. This is where the magic of arithmetic sequences comes into play, guys!

Identifying the Pattern

The series 1, 3, 5, ..., 99 is an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In our case, the common difference is 2 (3 - 1 = 2, 5 - 3 = 2, and so on). This constant difference is what makes it an arithmetic sequence and allows us to use specific formulas to find the sum.

Understanding the pattern is super crucial. We can see that each number is 2 more than the previous one. This consistent increase helps us predict the next numbers in the sequence and ultimately find the total number of terms. By identifying this arithmetic progression, we can use formulas designed for these types of series. It's like having a secret weapon in our math arsenal!

Determining the Number of Terms

Now that we know it's an arithmetic sequence, we need to find out how many terms are in the series. This is essential because the number of terms directly impacts the sum. To find the number of terms (n), we can use the formula for the nth term of an arithmetic sequence:

an = a1 + (n - 1)d

Where:

  • an is the nth term (in our case, 99)
  • a1 is the first term (which is 1)
  • n is the number of terms (what we want to find)
  • d is the common difference (which is 2)

Plugging in the values, we get:

99 = 1 + (n - 1)2

Let's solve for n:

99 = 1 + 2n - 2
99 = 2n - 1
100 = 2n
n = 50

So, there are 50 terms in the series. Awesome! Now we're one step closer to solving the whole thing. Finding the number of terms is like unlocking the next level in a game. We now have a crucial piece of information that will help us calculate the final sum. It's all about breaking down the problem into smaller, manageable steps, right?

Calculating the Sum Using the Arithmetic Series Formula

Now that we know the series is arithmetic and we've found the number of terms, we can use the formula for the sum of an arithmetic series. This formula is a real lifesaver, guys, as it makes calculating such sums a breeze. The formula is:

S = n/2 * (a1 + an)

Where:

  • S is the sum of the series
  • n is the number of terms (which we found to be 50)
  • a1 is the first term (which is 1)
  • an is the last term (which is 99)

Applying the Formula

Let's plug in the values we know:

S = 50/2 * (1 + 99)
S = 25 * 100
S = 2500

So, the sum of the series S = 1 + 3 + 5 + ... + 99 is 2500! How cool is that? By using the arithmetic series formula, we've managed to find the sum of 50 numbers without actually adding them all individually. This formula is a powerful tool, and it's something you can use in many similar problems. It's like having a superpower in math!

Step-by-Step Breakdown

To make sure we've got this down pat, let's quickly recap the steps we took:

  1. Identified the series as an arithmetic sequence with a common difference of 2.
  2. Determined the number of terms using the formula an = a1 + (n - 1)d, finding n = 50.
  3. Applied the arithmetic series formula S = n/2 * (a1 + an) to calculate the sum.
  4. Found the sum S = 2500.

Breaking down the problem into these steps makes it much easier to tackle. Each step builds upon the previous one, leading us to the final solution. It's like constructing a building, brick by brick, until you have a complete structure. And in this case, our structure is the answer to the problem!

Alternative Method: Pairing the Numbers

There's also another cool way to think about this problem, guys, which can give you a more intuitive understanding of why the formula works. Instead of just plugging numbers into a formula, this method helps you visualize the sum. We can pair the numbers in the series like this:

  • 1 + 99 = 100
  • 3 + 97 = 100
  • 5 + 95 = 100
  • ...

Visualizing the Sum

Notice that each pair adds up to 100. This is because we're pairing the smallest number with the largest, the second smallest with the second largest, and so on. The constant sum for each pair makes the calculation much simpler. It's like creating little bundles of 100s, which are super easy to add up.

How many pairs do we have? Since there are 50 terms in the series, we'll have 25 pairs (50 / 2 = 25). Each pair sums to 100, so the total sum is:

25 pairs * 100 = 2500

Why This Method Works

This method provides a visual and intuitive way to understand why the arithmetic series formula works. By pairing the numbers, we're essentially finding the average of the first and last terms (which is (1 + 99) / 2 = 50) and then multiplying it by the number of terms (50). This is exactly what the formula S = n/2 * (a1 + an) does!

Using this pairing method can be super helpful for those who prefer a more visual approach to math problems. It's like seeing the math in action, rather than just memorizing a formula. Plus, it's a great way to double-check your answer and make sure you're on the right track.

Real-World Applications of Arithmetic Series

So, you might be thinking,