Problem 51 - Beginning and Intermediate Algebra

Question

Evaluate each series. $$\sum_{i=3}^{6}\left(i^{2}\right)$$

Step-by-Step Solution

Answer

The series \( \sum_{i=3}^{6}\left(i^{2}\right) \) can be evaluated by calculating the sum of the squares of each term from 3 to 6: \(3^2 = 9\), \(4^2 = 16\), \(5^2 = 25\), and \(6^2 = 36\). Adding these together, we get \(9 + 16 + 25 + 36 = 86\). Therefore, the value of the series is 86.

Step 1: Identify the terms of the series

First, let's identify the terms of the series. We have: - Term 1: i = 3 - Term 2: i = 4 - Term 3: i = 5 - Term 4: i = 6

Step 2: Calculate the square of each term

Next, let's calculate the square of each term: - Term 1: \(3^2 = 9\) - Term 2: \(4^2 = 16\) - Term 3: \(5^2 = 25\) - Term 4: \(6^2 = 36\)

Step 3: Add the squared terms

Now that we have the square of each term, we can evaluate the series by adding them together: $$\sum_{i=3}^{6}\left(i^{2}\right) = 9 + 16 + 25 + 36$$

Step 4: Calculate the sum

Finally, let's add the squared terms together to find the value of the series: $$\sum_{i=3}^{6}\left(i^{2}\right) = 9 + 16 + 25 + 36 = 86$$ The value of the series is 86.

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In algebra, these symbols represent numbers in formulas and equations. It's like a language that lets us describe relationships between different quantities.

Algebra is essential for solving various problems in a structured way. It allows us to create equations that can be solved to find unknown values. In the context of our exercise, algebra helps us to understand and evaluate the series by representing terms with a variable.

• Variables: Symbols like \(i\) in our series which can take different values.
• Expressions: Combinations of variables and constants, such as \(i^2\), that we evaluate.
• Equations: Mathematical statements that assert the equality of two expressions.

Using algebraic principles, we systematically evaluate each term in the series and then sum these values to reach a solution.

Summation Notation

Summation notation is a compact way of representing the sum of a sequence of numbers. It is especially useful for working with series like the one we have. The Greek letter sigma (\(\sum\)) is used to indicate summation. This notation simplifies the process of writing and evaluating long sums. Let's break it down:

• Index of Summation: The variable \(i\) is called the index, representing the position of each term in the series.
• Lower and Upper Limits: The numbers 3 and 6 are the lower and upper limits, respectively. These define the range over which \(i\) will vary.
• Expression to be Summed: \(i^2\) is the expression applied to each index value.

The summation notation \(\sum_{i=3}^{6}\left(i^{2}\right)\) tells us to evaluate \(i^2\) for each integer \(i\) from 3 to 6 and then sum the results.

This systematic approach is valuable in various mathematical calculations, streamlining the computation and improving clarity.

Square Numbers

Square numbers are the product of an integer multiplied by itself. They are called "square" because they can represent the area of a square with sides of integer length. For example, \(4^2 = 16\) represents a square with each side length 4.

• A square number is written as \(n^2\), where \(n\) is an integer.
• Each squared term is calculated independently, as shown in our exercise.

For our series, we need to find the square of each number between 3 and 6:

• \(3^2 = 9\)
• \(4^2 = 16\)
• \(5^2 = 25\)
• \(6^2 = 36\)

Calculating square numbers is an important skill in algebra and is frequently used in series evaluation and other areas of mathematics. It helps us to appreciate the underlying patterns and relationships between numbers.