Problem 73 - A Graphical Approach to College Algebra

Question

Use summation notation to write each series. Start the index at \(i=1\). $$\frac{2}{5(1)}+\frac{2}{5(2)}+\frac{2}{5(3)}+\dots+\frac{2}{5(100)}$$

Step-by-Step Solution

Answer

\( \sum_{i=1}^{100} \frac{2}{5i} \)

Step 1: Identify Pattern

In the given series, each term is in the form of \(\frac{2}{5i}\) where \(i\) represents the position of the term in the series. Observe that the numerator is constant (2), while the denominator is a product of 5 and the position \(i\).

Step 2: Choose Summation Notation

The summation notation requires us to express the series \(\frac{2}{5(1)}+\frac{2}{5(2)}+\frac{2}{5(3)}+\cdots+\frac{2}{5(100)}\) compactly. Here, each term can be written as \(\frac{2}{5i}\). We start the index \(i\) at 1 and end at 100.

Step 3: Write in Summation Notation

Using the pattern identified, the series can be expressed in summation notation as \( \sum_{i=1}^{100} \frac{2}{5i} \). This compactly represents the sum of all terms from the first to the one hundredth, following the identified pattern.

Key Concepts

Series

In mathematics, a series is essentially the summation of a sequence of numbers. It involves adding up the terms of a sequence following a specific rule or pattern. For example, the series \( \frac{2}{5(1)}+\frac{2}{5(2)}+\frac{2}{5(3)}+\ldots+\frac{2}{5(100)} \) is an addition of different fractions that follow a particular rule. Each fraction can be seen as a part of the whole series, all governed by the same mathematical pattern. When we refer to a series, we typically talk about the sum from the first term to the last term, which can go on indefinitely or may end after a certain number of terms, as in this case from 1 to 100.

Mathematical Pattern

A mathematical pattern is a sequence that follows a specific, identifiable rule. In the present exercise, the pattern is visible in the repeating structure of each term: \( \frac{2}{5i} \).

• The numerator is always 2, showing no variation.
• The denominator follows a clear pattern, which is 5 multiplied by the index \( i \). This results in fractions like \( \frac{2}{5(1)} \), \( \frac{2}{5(2)} \), and so on.

Recognizing these patterns helps you predict what the next terms in the series will look like and assists in writing or simplifying them using summation notation.

Indexing

Indexing is the use of indices (or index variables) to specify terms' positions within a sequence or series. The variable \( i \) is such an index. In our series, \( i \) starts at 1 and ends at 100, noting the position of each term. This index

• tells us that the series begins with \( i = 1 \)
• and ends with \( i = 100 \)

Understanding indexing is crucial because it defines the boundaries of the series and allows you to accurately calculate how many terms are involved. This is especially useful in summation, where the range of the index indicates the starting and ending points of the series.

Notation

Mathematical notation is a set of written symbols used to represent numbers, functions, relationships, and concepts. Summation notation comes in handy for simplifying complex series and sums. In this context, it allows us to express the lengthy series \( \frac{2}{5(1)}+\frac{2}{5(2)}+\cdots+\frac{2}{5(100)} \) in a more concise form: \[ \sum_{i=1}^{100} \frac{2}{5i} \] This notation provides valuable information:

• The greek letter \( \sum \) signifies the sum of a series.
• The expression \( \frac{2}{5i} \) describes the general term.
• \( i=1 \) to \( 100 \) is the range of this summation, indicating the start and end of the index.

Using this notation simplifies reading, communicating, and calculating complex mathematical ideas efficiently.