Evaluate ∑ N = 1 12 ( 2 N + 5 \sum_{n=1}^{12} (2n+5 ∑ N = 1 12 ​ ( 2 N + 5 ]A. 29 B. 36 36 36 C. 216 D. 432

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Introduction

In mathematics, summation is a fundamental concept used to represent the sum of a series of numbers. It is denoted by the symbol $\sum$ and is used to calculate the total value of a sequence of numbers. In this article, we will evaluate the summation $\sum_{n=1}^{12} (2n+5)$, which represents the sum of the expression $(2n+5)$ for $n$ ranging from $1$ to $12$.

Understanding the Summation Notation

The summation notation $\sum_{n=1}^{12} (2n+5)$ can be broken down into two parts:

• The lower limit of the summation, which is $1$, indicates that the summation starts from $n=1$.
• The upper limit of the summation, which is $12$, indicates that the summation ends at $n=12$.
• The expression $(2n+5)$ is the term that is being summed.

Evaluating the Summation

To evaluate the summation, we need to substitute the values of $n$ from $1$ to $12$ into the expression $(2n+5)$ and calculate the sum.

Step 1: Substitute $n=1$ into the expression $(2n+5)$

When $n=1$, the expression becomes $(2(1)+5) = 7$.

Step 2: Substitute $n=2$ into the expression $(2n+5)$

When $n=2$, the expression becomes $(2(2)+5) = 9$.

Step 3: Substitute $n=3$ into the expression $(2n+5)$

When $n=3$, the expression becomes $(2(3)+5) = 11$.

Step 4: Substitute $n=4$ into the expression $(2n+5)$

When $n=4$, the expression becomes $(2(4)+5) = 13$.

Step 5: Substitute $n=5$ into the expression $(2n+5)$

When $n=5$, the expression becomes $(2(5)+5) = 15$.

Step 6: Substitute $n=6$ into the expression $(2n+5)$

When $n=6$, the expression becomes $(2(6)+5) = 17$.

Step 7: Substitute $n=7$ into the expression $(2n+5)$

When $n=7$, the expression becomes $(2(7)+5) = 19$.

Step 8: Substitute $n=8$ into the expression $(2n+5)$

When $n=8$, the expression becomes $(2(8)+5) = 21$.

Step 9: Substitute $n=9$ into the expression $(2n+5)$

When $n=9$, the expression becomes $(2(9)+5) = 23$.

Step 10: Substitute $n=10$ into the expression $(2n+5)$

When $n=10$, the expression becomes $(2(10)+5) = 25$.

Step 11: Substitute $n=11$ into the expression $(2n+5)$

When $n=11$, the expression becomes $(2(11)+5) = 27$.

Step 12: Substitute $n=12$ into the expression $(2n+5)$

When $n=12$, the expression becomes $(2(12)+5) = 29$.

Calculating the Sum

Now that we have calculated the value of the expression $(2n+5)$ for each value of $n$ from $1$ to $12$, we can calculate the sum:

$7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 = 216$

Conclusion

In this article, we evaluated the summation $\sum_{n=1}^{12} (2n+5)$, which represents the sum of the expression $(2n+5)$ for $n$ ranging from $1$ to $12$. We broke down the summation notation, substituted the values of $n$ into the expression, and calculated the sum. The final answer is $\boxed{216}$.

Final Answer

The final answer is $\boxed{216}$.

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Introduction

In our previous article, we evaluated the summation $\sum_{n=1}^{12} (2n+5)$, which represents the sum of the expression $(2n+5)$ for $n$ ranging from $1$ to $12$. In this article, we will answer some frequently asked questions related to the evaluation of this summation.

Q1: What is the formula for the summation $\sum_{n=1}^{12} (2n+5)$?

A1: The formula for the summation $\sum_{n=1}^{12} (2n+5)$ is $(2n+5)$, where $n$ ranges from $1$ to $12$.

Q2: How do I evaluate the summation $\sum_{n=1}^{12} (2n+5)$?

A2: To evaluate the summation $\sum_{n=1}^{12} (2n+5)$, you need to substitute the values of $n$ from $1$ to $12$ into the expression $(2n+5)$ and calculate the sum.

Q3: What is the value of the expression $(2n+5)$ when $n=1$?

A3: When $n=1$, the expression $(2n+5)$ becomes $(2(1)+5) = 7$.

Q4: What is the value of the expression $(2n+5)$ when $n=12$?

A4: When $n=12$, the expression $(2n+5)$ becomes $(2(12)+5) = 29$.

Q5: How do I calculate the sum of the expression $(2n+5)$ for $n$ ranging from $1$ to $12$?

A5: To calculate the sum of the expression $(2n+5)$ for $n$ ranging from $1$ to $12$, you need to add up the values of the expression for each value of $n$ from $1$ to $12$.

Q6: What is the final answer to the summation $\sum_{n=1}^{12} (2n+5)$?

A6: The final answer to the summation $\sum_{n=1}^{12} (2n+5)$ is $\boxed{216}$.

Q7: Can I use a formula to calculate the sum of the expression $(2n+5)$ for $n$ ranging from $1$ to $12$?

A7: Yes, you can use the formula for the sum of an arithmetic series to calculate the sum of the expression $(2n+5)$ for $n$ ranging from $1$ to $12$. The formula is:

$$\sum_{n=1}^{n} (2n+5) = \frac{n}{2} (a_1 + a_n)$$

where $a_1$ is the first term and $a_n$ is the last term.

Q8: How do I apply the formula for the sum of an arithmetic series to the summation $\sum_{n=1}^{12} (2n+5)$?

A8: To apply the formula for the sum of an arithmetic series to the summation $\sum_{n=1}^{12} (2n+5)$, you need to identify the first term and the last term of the series. The first term is $a_1 = (2(1)+5) = 7$ and the last term is $a_n = (2(12)+5) = 29$. Then, you can plug these values into the formula:

$$\sum_{n=1}^{12} (2n+5) = \frac{12}{2} (7 + 29)$$

$$\sum_{n=1}^{12} (2n+5) = 6 (36)$$

$$\sum_{n=1}^{12} (2n+5) = 216$$

Conclusion

In this article, we answered some frequently asked questions related to the evaluation of the summation $\sum_{n=1}^{12} (2n+5)$. We provided step-by-step solutions to each question and explained the concepts and formulas used to evaluate the summation.

Final Answer

The final answer to the summation $\sum_{n=1}^{12} (2n+5)$ is $\boxed{216}$.