What Is The Sum Of The Series? ∑ K = 1 4 3 K 2 \sum_{k=1}^4 3k^2 ∑ K = 1 4 3 K 2 Enter Your Answer In The Box. □ \square □

Mar 13, 2025 by ADMIN 125 views

$What is the Sum of the Series? $\sum_{k=1}^4 3k^2$$

Introduction

In mathematics, a series is a sequence of numbers that are added together. The sum of a series is the total value obtained by adding all the terms in the series. In this article, we will explore the sum of a specific series, $\sum_{k=1}^4 3k^2$ . This series involves a quadratic expression, and we will use algebraic techniques to find its sum.

Understanding the Series

The given series is $\sum_{k=1}^4 3k^2$ . This means that we need to find the sum of the terms $3k^2$ for $k$ ranging from 1 to 4. In other words, we need to evaluate the expression $3(1)^2 + 3(2)^2 + 3(3)^2 + 3(4)^2$ .

Evaluating the Series

To evaluate the series, we can start by substituting the values of $k$ into the expression $3k^2$ . For $k=1$ , we have $3(1)^2 = 3$ . For $k=2$ , we have $3(2)^2 = 12$ . For $k=3$ , we have $3(3)^2 = 27$ . Finally, for $k=4$ , we have $3(4)^2 = 48$ .

Finding the Sum

Now that we have evaluated the terms of the series, we can find the sum by adding them together. The sum is $3 + 12 + 27 + 48 = 90$ .

Conclusion

In this article, we explored the sum of the series $\sum_{k=1}^4 3k^2$ . We used algebraic techniques to evaluate the terms of the series and found the sum to be 90. This example illustrates the importance of understanding series and their sums in mathematics.

Algebraic Techniques for Evaluating Series

When evaluating a series, we can use various algebraic techniques to simplify the expression. In this case, we used the distributive property to expand the expression $3k^2$ and then added the terms together. Other algebraic techniques that can be used to evaluate series include factoring, combining like terms, and using the commutative and associative properties of addition.

Importance of Series in Mathematics

Series are an important concept in mathematics, and they have numerous applications in various fields. In calculus, series are used to represent functions as infinite sums of terms. In number theory, series are used to study the properties of integers and prime numbers. In statistics, series are used to model real-world data and make predictions.

Real-World Applications of Series

Series have numerous real-world applications in fields such as finance, economics, and engineering. In finance, series are used to model stock prices and predict future returns. In economics, series are used to study the behavior of economic systems and make predictions about future trends. In engineering, series are used to design and optimize systems such as electrical circuits and mechanical systems.

Conclusion

In conclusion, the sum of the series $\sum_{k=1}^4 3k^2$ is 90. This example illustrates the importance of understanding series and their sums in mathematics. Series have numerous applications in various fields, and they are an essential concept in mathematics.

Final Answer

The final answer is $\boxed{90}$ .

Introduction

In our previous article, we explored the sum of the series $\sum_{k=1}^4 3k^2$ . In this article, we will answer some frequently asked questions (FAQs) about series. These FAQs cover a range of topics, from the basics of series to more advanced concepts.

Q: What is a series in mathematics?

A: A series is a sequence of numbers that are added together. It is a way of representing a sum of terms, where each term is a number or an expression.

Q: What is the difference between a series and a sequence?

A: A sequence is a list of numbers in a specific order, whereas a series is the sum of the terms in a sequence. For example, the sequence $1, 2, 3, 4, 5$ is a list of numbers, whereas the series $\sum_{k=1}^5 k$ is the sum of the terms in the sequence.

Q: How do I evaluate a series?

A: To evaluate a series, you need to follow these steps:

Identify the terms in the series.
Evaluate each term separately.
Add the terms together to find the sum.

Q: What are some common types of series?

A: There are several types of series, including:

Arithmetic series: a series where each term is obtained by adding a fixed constant to the previous term.
Geometric series: a series where each term is obtained by multiplying the previous term by a fixed constant.
Harmonic series: a series where each term is the reciprocal of a positive integer.
Power series: a series where each term is a power of a variable.

Q: How do I find the sum of a series?

A: To find the sum of a series, you can use various algebraic techniques, such as:

Factoring: breaking down the series into simpler terms.
Combining like terms: adding or subtracting terms that have the same variable.
Using the distributive property: expanding the series by multiplying each term by a constant.

Q: What are some real-world applications of series?

A: Series have numerous real-world applications in fields such as:

Finance: modeling stock prices and predicting future returns.
Economics: studying the behavior of economic systems and making predictions about future trends.
Engineering: designing and optimizing systems such as electrical circuits and mechanical systems.

Q: How do I use series in calculus?

A: In calculus, series are used to represent functions as infinite sums of terms. You can use series to:

Approximate functions: using a series to approximate the value of a function.
Find derivatives: using a series to find the derivative of a function.
Find integrals: using a series to find the integral of a function.

Q: What are some common mistakes to avoid when working with series?

A: Some common mistakes to avoid when working with series include:

Not following the order of operations: evaluating the terms in the series in the correct order.
Not using the distributive property: expanding the series by multiplying each term by a constant.
Not combining like terms: adding or subtracting terms that have the same variable.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about series. We covered a range of topics, from the basics of series to more advanced concepts. By understanding series and their applications, you can solve a wide range of mathematical problems and make predictions about real-world phenomena.

Final Answer

The final answer is $\boxed{90}$ .