Find ∑ K = 1 30 ( − 7 + ( K − 1 ) ( 3 ) \sum_{k=1}^{30}(-7+(k-1)(3) ∑ K = 1 30 ​ ( − 7 + ( K − 1 ) ( 3 ) ].A. 87 \quad 87 87 B. 206 \quad 206 206 C. 780 \quad 780 780 D. 1 , 095 \quad 1,095 1 , 095 E. 1 , 200 \quad 1,200 1 , 200

Introduction

In this article, we will delve into the world of summation and explore a specific problem that requires us to find the sum of a given expression. The problem involves finding the sum of the expression $(-7+(k-1)(3))$ from $k=1$ to $k=30$. We will break down the solution into manageable steps and provide a clear explanation of each step.

Understanding the Problem

The given problem is a summation problem, which means we need to find the sum of a given expression over a specific range of values. In this case, the expression is $(-7+(k-1)(3))$, and we need to find the sum from $k=1$ to $k=30$. This means we will be evaluating the expression for each value of $k$ from 1 to 30 and then adding up the results.

Breaking Down the Expression

Before we can start evaluating the expression, we need to break it down into its individual components. The expression can be broken down into two parts: $-7$ and $(k-1)(3)$. The first part is a constant, while the second part is a linear expression that depends on the value of $k$.

Evaluating the Expression

Now that we have broken down the expression, we can start evaluating it for each value of $k$ from 1 to 30. We will use the formula for the expression: $(-7+(k-1)(3))$. We will substitute each value of $k$ into the formula and evaluate the expression.

Calculating the Sum

Once we have evaluated the expression for each value of $k$, we can start calculating the sum. We will add up the results of each evaluation to find the final sum.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

Step 1: Evaluate the Expression for Each Value of k

We will start by evaluating the expression for each value of $k$ from 1 to 30. We will use the formula: $(-7+(k-1)(3))$. We will substitute each value of $k$ into the formula and evaluate the expression.

Step 2: Calculate the Sum

Now that we have evaluated the expression for each value of $k$, we can start calculating the sum. We will add up the results of each evaluation to find the final sum.

The sum of the expression $(-7+(k-1)(3))$ from $k=1$ to $k=30$ is:

-7 + (-4) + (-1) + 2 + 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 + 35 + 38 + 41 + 44 + 47 + 50 + 53 + 56 + 59 + 62 + 65 + 68 + 71 + 74 + 77 + 80 = 206

Therefore, the final answer is $\boxed{206}$.

Conclusion

Q: What is the formula for the given summation problem?

A: The formula for the given summation problem is $(-7+(k-1)(3))$, where $k$ ranges from 1 to 30.

Q: How do I evaluate the expression for each value of k?

A: To evaluate the expression for each value of $k$, you need to substitute each value of $k$ into the formula: $(-7+(k-1)(3))$. Then, you need to calculate the result of the expression for each value of $k$.

Q: What is the sum of the expression (-7+(k-1)(3)) from k=1 to k=30?

A: The sum of the expression $(-7+(k-1)(3))$ from $k=1$ to $k=30$ is 206.

Q: Can I use a calculator to evaluate the expression for each value of k?

A: Yes, you can use a calculator to evaluate the expression for each value of $k$. However, it's always a good idea to double-check your calculations to ensure accuracy.

Q: What if I make a mistake in evaluating the expression for each value of k?

A: If you make a mistake in evaluating the expression for each value of $k$, you may end up with an incorrect sum. To avoid this, make sure to double-check your calculations and use a calculator to verify your results.

Q: Can I use a formula to find the sum of the expression (-7+(k-1)(3)) from k=1 to k=30?

A: Yes, you can use a formula to find the sum of the expression $(-7+(k-1)(3))$ from $k=1$ to $k=30$. The formula is:

$\sum_{k=1}^{30} (-7+(k-1)(3)) = (-7+(-4)+(-1)+2+5+8+11+14+17+20+23+26+29+32+35+38+41+44+47+50+53+56+59+62+65+68+71+74+77+80)$

Q: How do I simplify the formula for the sum of the expression (-7+(k-1)(3)) from k=1 to k=30?

A: To simplify the formula for the sum of the expression $(-7+(k-1)(3))$ from $k=1$ to $k=30$, you can use the following steps:

1. Combine like terms: $(-7+(-4)+(-1)+2+5+8+11+14+17+20+23+26+29+32+35+38+41+44+47+50+53+56+59+62+65+68+71+74+77+80)$
2. Simplify the expression: $(-7+(-4)+(-1)+2+5+8+11+14+17+20+23+26+29+32+35+38+41+44+47+50+53+56+59+62+65+68+71+74+77+80) = 206$

Q: What is the final answer to the summation problem?

A: The final answer to the summation problem is 206.

Conclusion

In this article, we have answered some frequently asked questions about the summation problem. We have provided step-by-step solutions to the problem and explained how to evaluate the expression for each value of $k$. We have also provided a formula to find the sum of the expression $(-7+(k-1)(3))$ from $k=1$ to $k=30$. The final answer to the summation problem is 206.