Understanding the Series Representation

Oct 13, 2025 by windailys.com 78 views

What is the Expanded Form of the Series Represented Below?

Understanding the Series Representation

The given series is represented as $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ . This is a summation notation, where the sum is taken over the values of $k$ from $1$ to $5$ . The expression inside the summation is $[-3 + (k-1) \cdot 5]$ . To find the expanded form of the series, we need to evaluate this expression for each value of $k$ from $1$ to $5$ .

Evaluating the Expression for Each Value of k

Let's start by evaluating the expression for $k=1$ .

$[-3 + (1-1) \cdot 5] = [-3 + 0 \cdot 5] = -3$

Next, let's evaluate the expression for $k=2$ .

$[-3 + (2-1) \cdot 5] = [-3 + 1 \cdot 5] = -3 + 5 = 2$

Now, let's evaluate the expression for $k=3$ .

$[-3 + (3-1) \cdot 5] = [-3 + 2 \cdot 5] = -3 + 10 = 7$

Moving on, let's evaluate the expression for $k=4$ .

$[-3 + (4-1) \cdot 5] = [-3 + 3 \cdot 5] = -3 + 15 = 12$

Finally, let's evaluate the expression for $k=5$ .

$[-3 + (5-1) \cdot 5] = [-3 + 4 \cdot 5] = -3 + 20 = 17$

Expanded Form of the Series

Now that we have evaluated the expression for each value of $k$ from $1$ to $5$ , we can write the expanded form of the series as:

$S_5 = -3 + 2 + 7 + 12 + 17$

Comparing with the Options

Let's compare the expanded form of the series with the options given:

A. $-3 + 2 + 7 + 12 + 17$ B. $-3 + 3 + 8 + 13 + 18$ C. $2 + 7 + 12 + 17 + 22$ D. $5 + 2 + (-1) + 12 + 17$

The expanded form of the series matches with option A.

Conclusion

In this article, we have evaluated the expression inside the summation notation for each value of $k$ from $1$ to $5$ and found the expanded form of the series. We have also compared the expanded form with the options given and found that it matches with option A.
Frequently Asked Questions (FAQs) About the Series Representation

Q: What is the series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ ?

A: The series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ is a summation notation, where the sum is taken over the values of $k$ from $1$ to $5$ . The expression inside the summation is $[-3 + (k-1) \cdot 5]$ .

Q: How do I evaluate the expression inside the summation notation?

A: To evaluate the expression inside the summation notation, you need to substitute the value of $k$ into the expression and simplify. For example, for $k=1$ , the expression becomes $[-3 + (1-1) \cdot 5] = [-3 + 0 \cdot 5] = -3$ .

Q: What is the expanded form of the series $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ ?

A: The expanded form of the series $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ is $-3 + 2 + 7 + 12 + 17$ .

Q: How do I compare the expanded form of the series with the options given?

A: To compare the expanded form of the series with the options given, you need to match the terms of the expanded form with the terms of the options. In this case, the expanded form $-3 + 2 + 7 + 12 + 17$ matches with option A.

Q: What is the significance of the series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ ?

A: The series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ is a mathematical expression that can be used to model real-world problems. For example, it can be used to model the cost of producing a certain number of items, where the cost of each item increases by a certain amount for each additional item produced.

Q: How do I use the series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ in real-world problems?

A: To use the series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ in real-world problems, you need to substitute the values of the variables into the expression and simplify. For example, if you want to model the cost of producing 5 items, where the cost of each item increases by $5 for each additional item produced, you can substitute $k=1, 2, 3, 4, 5$ into the expression and simplify.

Q: What are some common applications of the series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ ?

A: Some common applications of the series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ include:

Modeling the cost of producing a certain number of items
Modeling the revenue generated by selling a certain number of items
Modeling the profit made by selling a certain number of items
Modeling the cost of producing a certain number of items with varying costs

Q: How do I determine the value of the series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ ?

A: To determine the value of the series representation $S_5 = \sum_{k=1}^5 [-3 + (k-1) \cdot 5]$ , you need to evaluate the expression inside the summation notation for each value of $k$ from $1$ to $5$ and add up the results. In this case, the value of the series representation is $-3 + 2 + 7 + 12 + 17 = 25$ .