Problem 48 - Algebra and Trigonometry

Question

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 1+2+3+\dots+40 $$

Step-by-Step Solution

Answer

The sum \(1+2+3+...+40\) can be represented using summation notation as \(\Sigma_{i=1}^{40} i\).

Step 1: Understanding Summation Notation

Summation notation is a method used to denote the addition of a sequence of numbers. It uses the Greek letter sigma, denoted as \(\Sigma\), to indicate a sum. A number below sigma (\(\Sigma\)) represents the lower limit (the starting point of the sum), while a number above \(\Sigma\) represents the upper limit (where the sum ends). The variable to the right of sigma is what we add up. In this case, we're told to use i for the index of summation.

Step 2: Applying Summation Notation to the Given Sum

The sum \(1+2+3+...+40\) can be represented in summation notation as \(\Sigma_{i=1}^{40} i\) where i is the index and the summation runs from 1 to 40. We take each integer from 1 to 40, substitute it for i, and then add up all those results.

Key Concepts

Sequence of Numbers

When we talk about a sequence of numbers, we refer to an ordered list of numbers that follow a specific pattern or rule. In the example of the exercise, the sequence begins at 1 and ends at 40, increasing by 1 each time. This is a simple arithmetic sequence where each term increases by a constant, known as the common difference, which is 1 in this case.
Sequence of numbers are core elements in mathematics because they allow us to make predictions and recognize patterns. We can analyze sequences by breaking them down into components such as:

• Terms: Each individual number in the list.
• First Term: The starting number of a sequence, which is 1 in this example.
• Last Term: The final number in the list, which here is 40.
• Common Difference: The amount the sequence increases or decreases between terms, which is consistent across the entire sequence.

Understanding the structure of sequences of numbers is the foundation for using more complex mathematical tools, such as summation notation.

Index of Summation

The index of summation is a variable, often denoted by letters like 'i', 'j', or 'k', which indicates each term in the sequence that we're summing. In mathematical notation, it acts as a placeholder to represent each step in the sum.
When we look at summation notation \(\Sigma_{i=1}^{40} i\), the index of summation, 'i', represents each number from 1 to 40 in our sequence. The lower limit is 1, indicating where we start, and the upper limit is 40, showing where we stop the summation.
The index of summation is essential as it:

• Demarcates the start and end points for our add-up.
• Provides a clear method to substitute each term one by one and calculate the entire sum.
• Allows us to systematically express and compute sums for potentially large sets of numbers.

Every step taken by the index adds a new number from the sequence to the total, ensuring the operation is completed in an orderly manner. This helps in simplifying, analyzing, and computing lengthy numerical series efficiently.

Greek Letter Sigma

The Greek letter sigma, \(\Sigma\), is a powerful symbol used in mathematics to express the concept of summation. It serves as a compact way to indicate that a series of numbers should be summed together.
In the context of the exercise, the sigma notation \(\Sigma_{i=1}^{40} i\) represents the sum of the sequence of numbers from 1 to 40. It's the symbolic instruction to go through each number, from the lower limit at 1 to the upper limit at 40, using 'i' as the index.
Here are some key aspects of sigma notation:

• It holds a lower limit and an upper limit, marking the range of the sequence to be added.
• Understood universally as an indication of summation in mathematics, bypassing the need for lengthy expressions.
• Conveys succinctness and clarity, especially when handling sequences that extend over large ranges.

The sigma symbol simplifies complex mathematical expressions, making them easier to understand, communicate, and calculate effectively.