Problem 50 - College Algebra

Question

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$\frac{1}{3}+\frac{2}{4}+\frac{3}{5}+\dots+\frac{16}{16+2}$$

Step-by-Step Solution

Answer

\(\sum_{i=1}^{16}\frac{i}{i + 2}\)

Step 1: Identify the Pattern

The first step in solving this problem is to identify the pattern in the series. Each term in the given series is in the form \(\frac{i}{i+2}\), where i takes successive integer values starting from 1.

Step 2: Apply Summation Notation

After recognizing this pattern, we can express the series using summation notation. The index of summation starts from 1 and ends at 16. Therefore, the sum can be rewritten as: \(\sum_{i=1}^{16}\frac{i}{i + 2}\)

Key Concepts

Sigma Notation

Sigma notation, also known as summation notation, is a convenient way to express long sums in a compact form. It is represented using the Greek letter sigma, \( \Sigma \). This notation includes an expression for the terms of the series, the index of summation (often \( i \), \( j \), or \( k \) ), and the upper and lower bounds which indicate where the sum begins and ends.

For example, the sum \(\frac{1}{3}+\frac{2}{4}+\frac{3}{5}+\dots+\frac{16}{18}\) can be written in sigma notation as \(\sum_{i=1}^{16}\frac{i}{i + 2}\), where \(i\) is the index that starts at 1 and goes to 16. Such notation simplifies the representation of series and allows for easier manipulation in algebraic calculations.

Arithmetical Series

An arithmetical series is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term. The series \(1, 2, 3, ...\) is a simple example, with a common difference of 1.

In the given exercise, the terms are not in an arithmetical series, since the difference between them does not remain constant. The terms \(\frac{1}{3}, \frac{2}{4}, \frac{3}{5}, ...\) increase in a pattern, but by fractions that differ at each step, which means a different approach is needed for their summation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. For instance, the expression \(\frac{i}{i+2}\) seen in the exercise is an algebraic expression representing the terms of the given series. Such expressions are fundamental in algebra since they represent general values and can be manipulated according to algebra's rules to simplify or solve equations and series.

Understanding how to work with algebraic expressions is critical when using sigma notation because it often involves summing terms that can be variable.

Mathematical Patterns

Recognizing mathematical patterns is key to understanding and solving problems in mathematics. In the context of series and sequences, a pattern refers to a regularity or a predictable form that terms follow. For example, when we look at the series in the exercise, we may spot that each term has a numerator that increases by 1 and a denominator that is 2 more than the numerator.

These patterns often drive the way we can express series in more simplified forms like summation notation. By identifying the pattern \(\frac{i}{i+2}\) in the exercise, we can summarize and manipulate the entire series using algebraic principles without writing out each term explicitly. This is incredibly useful, especially for handling large sums or when analyzing the behavior of sequences and series.