Question

Evaluate expression. $$\sum_{n=1}^{5} n$$

Short answer

The sum of the numbers from 1 to 5 is 15.

Step-by-step solution

Understand the Sigma Notation

The sigma notation \(\sum_{n=1}^{5} n\) represents the sum of the values of the variable \(n\) as it changes from 1 to 5.

List the Terms

List all the terms represented by the sigma notation: 1, 2, 3, 4, 5.

Perform the Summation

Add the listed numbers to find the sum: \(1 + 2 + 3 + 4 + 5\).

Calculate the Result

Calculate the sum: \(1 + 2 + 3 + 4 + 5 = 15\).

Key concepts

Summation Process

The summation process is a fundamental mathematical concept used to find the total sum of a sequence of numbers. This process starts with identifying the sequence of elements to sum and then simply adding them together to find a cumulative total.

To understand the summation process, one must first be able to recognize the elements that are being summed. In our example, the elements are the natural numbers from 1 to 5. The process involves taking each number in sequence and adding it to the sum of all preceding numbers until all the elements have been included. These step-by-step additions lead us to the final result.

An important aspect of the summation process is that it is not restricted to simple sequences or small numbers. It can be used for intricate sequences and large collections of numbers, provided that there is a clear rule determining which numbers to include in the sum.

In the context of our exercise, the summation process involved adding the consecutive integers from 1 to 5, which resulted in a sum of 15.

Sigma Notation

Sigma notation is a concise and efficient way to represent the summation of a series of numbers. The Greek uppercase letter sigma \( \Sigma \) is used to denote the sum of a sequence. This notation includes an expression for the terms to be summed, the variable that represents the members of the series, the lower bound, and the upper bound of the summation.

For example, \( \sum_{n=1}^{5} n \) indicates that the variable \( n \) starts at 1 and increases by 1 until it reaches 5. Each value of \( n \) should be substituted into the expression to the right of the sigma, which, in this case, is simply \( n \) itself.

Using sigma notation provides many benefits. It streamlines the representation of the sum, especially with longer series or more complex terms. It also offers a clear and universally understood mathematical language that scholars can use to communicate complex summation problems and solutions.

Series and Sequences

In mathematics, it's crucial to understand the distinction and relationship between series and sequences. A sequence is a set of numbers arranged in a particular order following a specific rule. For instance, the sequence of positive integers \( 1, 2, 3, 4, 5 \) follows the rule that each number is one more than the previous number.

A series, on the other hand, is what you get when you sum the elements of a sequence. When we speak of the sum of a sequence, we are referring to a series. In the example of the sequence provided, the corresponding series would be the sum of these numbers \( 1+2+3+4+5 \) which yields 15.

Both concepts are interconnected and are fundamental in various areas of mathematics, including calculus and number theory. They are also instrumental in understanding real-world phenomena, such as financial calculations involving interest and the analysis of patterns within data sets.