Problem 5 - Algebra for College Students

Question

Find the sum of each series. $$\sum_{i=1}^{5} i$$

Step-by-Step Solution

Answer

15

Step 1 - Understand the Series

Identify the series that needs to be summed. Here, we have the series defined by the summation notation \(\sum_{i=1}^{5} i\).This notation means adding all integers from 1 to 5.

Step 2 - Expand the Series

Write out the terms of the series: \(1, 2, 3, 4, 5\)

Step 3 - Sum the Terms

Add the individual terms of the series: \(1 + 2 + 3 + 4 + 5\)

Step 4 - Calculate the Sum

Perform the addition: \(1 + 2 = 3\), \(3 + 3 = 6\), \(6 + 4 = 10\), \(10 + 5 = 15\)

Key Concepts

Series

In mathematics, a series is the sum of the terms of a sequence. For example, in the exercise, we have a simple series where we sum numbers from 1 to 5. A series can be finite or infinite, depending on whether it has a limited number of terms or continues indefinitely. Finite series, like the one in the exercise, are often easier to handle because they involve a specific endpoint. Understanding the structure of a series is crucial for solving summation problems effectively.

Basic Arithmetic

Basic arithmetic is the foundation of all mathematics. It includes operations like addition, subtraction, multiplication, and division. In the provided exercise, addition is used to find the sum of the series. Here are the steps broken down:

• First, list the terms: 1, 2, 3, 4, and 5.
• Then, add them one by one: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15.

By following these steps, you can clearly see how arithmetic operations are applied in the summation.

Summation

Summation is the operation of adding a sequence of numbers. The notation \( \sum_{i=1}^{5} i \) tells us to sum all integers from 1 to 5. This is a way of expressing a series compactly. The expanded form of this notation would be 1 + 2 + 3 + 4 + 5.
Summation notation is especially useful for expressing the addition of long or complex series without listing all terms. Learning to read and interpret summation notation helps streamline many mathematical processes. Understanding both the notation and the calculation process is key to mastering concepts involving sums of series.