Sum Of Even Numbers Between 71 And 149: A Step-by-Step Guide

by Mei Lin 61 views

Finding the sum of even numbers within a given range is a common mathematical problem that can be approached in a systematic way. This article will guide you through the process of calculating the sum of all even numbers between 71 and 149. We will break down the problem into manageable steps, making it easy to understand and solve. Whether you're a student learning about arithmetic series or simply someone looking to brush up on your math skills, this guide will provide you with a clear and concise method.

Understanding the Problem

Before we dive into the calculations, let's make sure we fully understand the problem. We need to find the sum of all even numbers that fall between 71 and 149. This means we're looking for the sum of numbers like 72, 74, 76, and so on, up to 148. The key here is to identify the first and last even numbers in the range and then use a formula to calculate the sum efficiently. Understanding the problem clearly is the first step towards solving it accurately. We need to be meticulous in identifying the even numbers within the specified range and ensure that we include all of them in our calculation. Now, let's move on to the next step: identifying the first and last even numbers in the given range. This will lay the groundwork for applying the arithmetic series formula, which will help us find the sum of all the even numbers. Remember, accuracy in problem definition leads to accuracy in the solution, so let's get this right from the start.

Identifying the First and Last Even Numbers

The next step in solving this problem is to pinpoint the first and last even numbers within the range of 71 to 149. This is crucial because these two numbers will define the boundaries of our series and help us determine the number of terms we need to sum. Let's start by identifying the first even number. Since 71 is an odd number, the next even number is 72. So, 72 is the first even number in our range. Now, let's find the last even number. 149 is also an odd number, so we need to look at the even number just before it, which is 148. Therefore, 148 is the last even number in our range. Now that we have identified the first (72) and last (148) even numbers, we can proceed to determine the number of terms in this series. This is a critical step because the number of terms directly impacts the sum. We will use the arithmetic sequence formula to figure this out, ensuring we have an accurate count. Getting these boundary numbers right is essential for the rest of the calculation, so let's make sure we've got them correct before moving on.

Determining the Number of Terms

Now that we know the first (72) and last (148) even numbers in our range, the next step is to determine the number of terms in this 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, since we are dealing with even numbers. To find the number of terms, 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 last term, a_1 is the first term, n is the number of terms, and d is the common difference. In our problem, a_n = 148, a_1 = 72, and d = 2. Plugging these values into the formula, we get: 148 = 72 + (n - 1)2. Now, we need to solve for n. First, subtract 72 from both sides: 76 = (n - 1)2. Next, divide both sides by 2: 38 = n - 1. Finally, add 1 to both sides: n = 39. So, there are 39 even numbers between 71 and 149. Knowing the number of terms is crucial for calculating the sum of the series accurately. This step ensures we include all even numbers in our calculation, without missing any or counting any twice. With this information in hand, we can now confidently move on to the final step: calculating the sum of the series using the arithmetic series formula. This will give us the final answer to our problem.

Calculating the Sum

With the number of terms determined, we can now proceed to calculate the sum of all even numbers between 71 and 149. To do this, we'll use the 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, a_1 is the first term, and a_n is the last term. We already know that n = 39, a_1 = 72, and a_n = 148. Plugging these values into the formula, we get: S_39 = 39/2 * (72 + 148). Now, let's simplify the equation. First, add 72 and 148: 72 + 148 = 220. Next, multiply 39/2 by 220: S_39 = 39/2 * 220. To make the calculation easier, we can first divide 220 by 2: 220 / 2 = 110. Now, multiply 39 by 110: 39 * 110 = 4290. Therefore, the sum of all even numbers between 71 and 149 is 4290. This final calculation brings together all the previous steps, giving us the answer we were looking for. By using the arithmetic series formula, we've efficiently calculated the sum without having to add each individual number. This method is not only accurate but also saves time, especially when dealing with larger ranges. Now that we have the final answer, let's take a moment to review the entire process to ensure we understand each step and can apply it to similar problems in the future.

Conclusion

In conclusion, to find the sum of all even numbers between 71 and 149, we followed a step-by-step process. First, we identified the first and last even numbers in the range, which were 72 and 148, respectively. Then, we determined the number of terms in the series using the arithmetic sequence formula, finding that there were 39 terms. Finally, we used the arithmetic series formula to calculate the sum, which resulted in 4290. This systematic approach not only helped us solve the problem accurately but also provided a clear method that can be applied to similar problems involving arithmetic series. Understanding each step and the underlying formulas is crucial for mastering these types of mathematical challenges. Remember, breaking down a complex problem into smaller, manageable steps makes it easier to solve. By identifying the key components, such as the first and last terms and the number of terms, we can apply the appropriate formulas and arrive at the correct solution. This exercise demonstrates the power of arithmetic sequences and series in simplifying calculations involving a series of numbers with a common difference. We hope this guide has been helpful in understanding how to approach and solve such problems effectively.