Solving the First Summation

Oct 13, 2025 by miodeya.com 133 views

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Introduction

In this article, we will delve into the world of mathematics and explore the concept of summation. Specifically, we will focus on finding the numerical answer to the given summation: $\sum_{n=1}^{81}(3n+7)$ . This problem requires a deep understanding of algebraic manipulation and the properties of summation.

Understanding Summation

Summation is a mathematical operation that involves adding up a series of numbers. It is denoted by the symbol $\sum$ and is used to represent the sum of a sequence of numbers. The general form of a summation is: $\sum_{i=a}^{b}f(i)$ , where $a$ and $b$ are the lower and upper limits of the summation, and $f(i)$ is the function being summed.

Breaking Down the Problem

To solve the given summation, we need to break it down into smaller, more manageable parts. We can start by expanding the summation using the distributive property: $\sum_{n=1}^{81}(3n+7) = \sum_{n=1}^{81}3n + \sum_{n=1}^{81}7$ .

Solving the First Summation

The first summation, $\sum_{n=1}^{81}3n$ , can be solved using the formula for the sum of an arithmetic series: $\sum_{n=1}^{n}an = \frac{n(a_1 + a_n)}{2}$ . In this case, $a = 3$ and $n = 81$ . Plugging in these values, we get: $\sum_{n=1}^{81}3n = \frac{81(3 + 3 \cdot 81)}{2}$ .

Simplifying the Expression

Simplifying the expression, we get: $\sum_{n=1}^{81}3n = \frac{81(3 + 243)}{2} = \frac{81 \cdot 246}{2} = 99 \cdot 246 = 24354$ .

Solving the Second Summation

The second summation, $\sum_{n=1}^{81}7$ , can be solved by multiplying the constant term by the number of terms: $\sum_{n=1}^{81}7 = 7 \cdot 81 = 567$ .

Combining the Results

Now that we have solved both summations, we can combine the results to find the final answer: $\sum_{n=1}^{81}(3n+7) = \sum_{n=1}^{81}3n + \sum_{n=1}^{81}7 = 24354 + 567 = 24921$ .

Conclusion

In this article, we have solved the given summation using algebraic manipulation and the properties of summation. We broke down the problem into smaller parts, solved each part separately, and then combined the results to find the final answer. The numerical answer to the summation is $\boxed{24921}$ .

Final Answer

The final answer is $\boxed{24921}$ .

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Introduction

In our previous article, we solved the summation $\sum_{n=1}^{81}(3n+7)$ and found the numerical answer to be $\boxed{24921}$ . In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is summation?

A: Summation is a mathematical operation that involves adding up a series of numbers. It is denoted by the symbol $\sum$ and is used to represent the sum of a sequence of numbers.

Q: How do I break down a summation?

A: To break down a summation, you can use the distributive property to expand the summation into smaller, more manageable parts. For example, $\sum_{n=1}^{81}(3n+7) = \sum_{n=1}^{81}3n + \sum_{n=1}^{81}7$ .

Q: How do I solve the first summation?

A: The first summation, $\sum_{n=1}^{81}3n$ , can be solved using the formula for the sum of an arithmetic series: $\sum_{n=1}^{n}an = \frac{n(a_1 + a_n)}{2}$ . In this case, $a = 3$ and $n = 81$ . Plugging in these values, we get: $\sum_{n=1}^{81}3n = \frac{81(3 + 3 \cdot 81)}{2}$ .

Q: How do I simplify the expression?

A: To simplify the expression, you can multiply the numbers inside the parentheses and then divide by 2. For example, $\sum_{n=1}^{81}3n = \frac{81(3 + 243)}{2} = \frac{81 \cdot 246}{2} = 99 \cdot 246 = 24354$ .

Q: How do I solve the second summation?

A: The second summation, $\sum_{n=1}^{81}7$ , can be solved by multiplying the constant term by the number of terms: $\sum_{n=1}^{81}7 = 7 \cdot 81 = 567$ .

Q: How do I combine the results?

A: To combine the results, you can add the two summations together: $\sum_{n=1}^{81}(3n+7) = \sum_{n=1}^{81}3n + \sum_{n=1}^{81}7 = 24354 + 567 = 24921$ .

Q: What is the final answer?

A: The final answer to the summation $\sum_{n=1}^{81}(3n+7)$ is $\boxed{24921}$ .

Q: Can I use this method to solve other summations?

A: Yes, you can use this method to solve other summations by breaking them down into smaller parts, solving each part separately, and then combining the results.

Q: What are some common mistakes to avoid when solving summations?

A: Some common mistakes to avoid when solving summations include:

Not breaking down the summation into smaller parts
Not using the distributive property to expand the summation
Not using the formula for the sum of an arithmetic series
Not simplifying the expression
Not combining the results correctly

Conclusion

In this article, we have provided a Q&A guide to help you understand the solution to the summation $\sum_{n=1}^{81}(3n+7)$ . We have covered topics such as breaking down the summation, solving the first and second summations, combining the results, and avoiding common mistakes. We hope this guide has been helpful in understanding the solution and answering any questions you may have.