Solving The Series: 4+8+12+...+400-3-6-9-...-300

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Hey guys! Let's break down this mathematical problem step by step. We've got a series that looks a bit intimidating at first glance, but don't worry, we'll make it super easy to understand. Our main goal here is to solve the series 4+8+12+...+400-3-6-9-...-300. This involves identifying patterns, using formulas for arithmetic series, and simplifying the expression. So, grab your thinking caps, and let’s dive in!

Understanding the Series

First off, let’s take a closer look at what we’re dealing with. We have two arithmetic series here: one where we're adding numbers and another where we're subtracting them. Let’s identify each one separately. The importance of understanding the series cannot be overstated; it’s the foundation upon which we build our solution.

The first part of our expression is 4+8+12+...+400. This is an arithmetic series where each term increases by 4. So, we start with 4, then add 4 to get 8, add another 4 to get 12, and so on, until we reach 400. Recognizing this pattern is crucial for applying the correct formulas later on.

Now, let’s look at the second part: -3-6-9-...-300. This is also an arithmetic series, but this time, we're subtracting. Each term decreases by 3. We start with -3, then subtract 3 to get -6, subtract another 3 to get -9, and continue until we reach -300. Notice the negative signs, as they’ll be important when we sum this series.

Breaking Down the First Series: 4+8+12+...+400

To tackle this, we need to find out a few key things about the first series. The key metrics to calculate for the first series include the number of terms, the common difference, and the sum. Let's get started!

Identifying the Common Difference and the Number of Terms

The common difference (d) in an arithmetic series is the constant amount by which each term increases. In this case, the common difference is 4. We're adding 4 each time: 4, 8, 12, and so on. Identifying the common difference is a fundamental step in working with arithmetic series.

Next, we need to figure out how many terms are in this series. To do this, we 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, 400)
  • a_1 is the first term (in our case, 4)
  • n is the number of terms (what we want to find)
  • d is the common difference (which is 4)

Plugging in our values, we get:

400 = 4 + (n - 1)4

Let's solve for n:

400 = 4 + 4n - 4 400 = 4n n = 100

So, there are 100 terms in the first series. Calculating the number of terms is essential for finding the sum of the series.

Calculating the Sum of the First Series

Now that we know there are 100 terms, we can find the sum of the series. The formula for the sum of an arithmetic series is:

S_n = n/2 * (a_1 + a_n)

Where:

  • S_n is the sum of the series
  • n is the number of terms (100)
  • a_1 is the first term (4)
  • a_n is the last term (400)

Let’s plug in the values:

S_100 = 100/2 * (4 + 400) S_100 = 50 * 404 S_100 = 20200

So, the sum of the first series (4+8+12+...+400) is 20200. Calculating the sum is the final step in understanding the first part of our problem.

Breaking Down the Second Series: -3-6-9-...-300

Alright, now let’s tackle the second series: -3-6-9-...-300. We'll follow the same steps as before, identifying the common difference, the number of terms, and then calculating the sum. Analyzing the second series involves similar steps but with negative numbers, which can add a bit of complexity. Let’s break it down.

Identifying the Common Difference and the Number of Terms

In this series, the common difference (d) is -3. We’re subtracting 3 each time: -3, -6, -9, and so on. The common difference here is negative, which indicates that the series is decreasing. Keeping track of the sign is crucial for accurate calculations.

To find the number of terms, we again 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, -300)
  • 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 (which is -3)

Plugging in our values, we get:

-300 = -3 + (n - 1)(-3)

Let's solve for n:

-300 = -3 - 3n + 3 -300 = -3n n = 100

So, there are also 100 terms in the second series. Interestingly, both series have the same number of terms. This might simplify our final calculation. Determining the number of terms is essential for summing the series correctly.

Calculating the Sum of the Second Series

Now that we know there are 100 terms, we can find the sum of the second series. We use the same formula for the sum of an arithmetic series:

S_n = n/2 * (a_1 + a_n)

Where:

  • S_n is the sum of the series
  • n is the number of terms (100)
  • a_1 is the first term (-3)
  • a_n is the last term (-300)

Let’s plug in the values:

S_100 = 100/2 * (-3 + (-300)) S_100 = 50 * (-303) S_100 = -15150

So, the sum of the second series (-3-6-9-...-300) is -15150. Calculating the sum completes our analysis of the second part of the problem.

Combining the Results

We've done the hard work of breaking down each series and finding their sums. Now, all that’s left is to combine these results to get our final answer. Combining the results involves simple addition of the two sums we calculated earlier. Let’s see how it’s done.

We found that the sum of the first series (4+8+12+...+400) is 20200, and the sum of the second series (-3-6-9-...-300) is -15150. To get the final answer, we simply add these two sums together:

Final Sum = 20200 + (-15150) Final Sum = 20200 - 15150 Final Sum = 5050

So, the final answer to the expression 4+8+12+...+400-3-6-9-...-300 is 5050. Congratulations, guys! We've successfully solved a complex mathematical problem by breaking it down into smaller, manageable parts. This approach of combining the results is a powerful technique in problem-solving.

Conclusion

In summary, solving the series 4+8+12+...+400-3-6-9-...-300 involved several key steps. We first identified the two arithmetic series, then found the common difference and the number of terms for each. We calculated the sums of each series separately and finally combined them to get the final result. The final answer is 5050. Remember, guys, breaking down complex problems into smaller steps makes them much easier to handle. Keep practicing, and you'll become math wizards in no time!