Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, express each description of a sum using summation notation. The sum from of \(n=0\) to \(n=4\) of \(5 n\)

Step-by-Step Solution

Answer

The sum is \( \sum_{n=0}^{4} 5n \).

Step 1: Understanding the Problem

The exercise provides a sum from \(n=0\) to \(n=4\) of the expression \(5n\). We need to represent this sum using summation notation.

Step 2: Setting up the Summation Notation

Summation notation is denoted by the Greek letter sigma \(\Sigma\). It includes the index of summation (which starts at a certain value and goes up to the upper limit) and the expression to be summed. For this problem, the expression is \(5n\), the index \(n\) starts at 0, and ends at 4.

Step 3: Writing the Sum in Summation Notation

We place the expression \(5n\) after the sigma symbol. Below the sigma, we write the starting index \(n = 0\), and above the sigma, we write the ending index \(n = 4\). Hence, the sum in summation notation is:\[ \sum_{n=0}^{4} 5n \]

Key Concepts

Sum Representation

When we talk about sum representation, we are discussing how a sum is expressed using mathematical symbols. In mathematics, especially in series and sequences, expressing sums in a compact form is crucial. This is where summation notation comes into play.
The problem given involves finding the sum of terms from a starting index to an ending index. The formula for each term in this case is given by the expression \(5n\). Instead of writing out each term, like \(5 \times 0, 5 \times 1, 5 \times 2,\) up to \(5 \times 4\), summation notation allows us to precisely package this into a concise form.
This form helps us quickly understand the range of values (or the bounds) and the condition of the index involved, importantly reducing room for error when dealing with longer sequences or more complex expressions.

Mathematical Notation

Mathematical notation refers to the system of symbols used to express mathematical ideas succinctly. Summation notation is a part of this system and helps simplify the writing of repeated addition. It allows mathematicians and professionals to convey ideas efficiently using universally understood symbols.
The sigma symbol \(\Sigma\) is at the heart of summation notation. This uppercase Greek letter denotes sum and is accompanied by limits specifying the starting and ending indices of the sum. Above the \(\Sigma\), we place the upper limit of the index, then below, we note the lower limit. Right next to the sigma, we show the expression to be summed, providing a full picture of the operation.
In our example, the notation \(\sum_{n=0}^{4} 5n\) succinctly instructs us to calculate the sum of \(5n\) as \(n\) varies from 0 to 4. Mathematical notation like this is essential because it conveys complex mathematical operations in an understandable and standardized form.

Index of Summation

The index of summation, in the context of summation notation, refers to the variable that takes on successive integer values from a starting point to an ending point as specified in the sum notation. This is a crucial part of understanding summation notation as it determines the actual values included in the sum.
In our exercise, the index \(n\) starts from 0 and rises to 4, which means it takes on the values 0, 1, 2, 3, and 4. The expression \(5n\) is evaluated for each of these values of \(n\). Each time the index takes a new value, the corresponding term \(5n\) is computed and added to the previous terms.
This approach enables us to systematically evaluate the components of the sum, offering clarity and structure. Understanding how the index of summation works is vital, as it dictates the number of terms evaluated and summed, shaping the outcome of the sum notation.