Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Evaluate the terms of each sum, where \(x_{1}=-2, x_{2}=-1, x_{3}=0, x_{4}=1,\) and \(x_{5}=2\) $$\sum_{i=1}^{5} \frac{x_{i}}{x_{i}+3}$$

Step-by-Step Solution

Answer

The sum is \(-\frac{37}{20}\).

Step 1: Identify the Components of the Sum

The problem provides the terms of a sum: \(x_1 = -2\), \(x_2 = -1\), \(x_3 = 0\), \(x_4 = 1\), and \(x_5 = 2\). We need to evaluate the expression \(\sum_{i=1}^{5} \frac{x_i}{x_i+3}\). This means we will substitute each value of \(x_i\) one by one and find the respective value of \(\frac{x_i}{x_i+3}\) for each.

Step 2: Substitute and Calculate Individual Terms

Substitute each \(x_i\) into the expression \(\frac{x_i}{x_i+3}\):- For \(x_1 = -2\), \(\frac{-2}{-2+3} = \frac{-2}{1} = -2\).- For \(x_2 = -1\), \(\frac{-1}{-1+3} = \frac{-1}{2}\).- For \(x_3 = 0\), \(\frac{0}{0+3} = 0\).- For \(x_4 = 1\), \(\frac{1}{1+3} = \frac{1}{4}\).- For \(x_5 = 2\), \(\frac{2}{2+3} = \frac{2}{5}\).

Step 3: Sum the Evaluated Terms

Next, we add up all the calculated fractions: \[-2 + \frac{-1}{2} + 0 + \frac{1}{4} + \frac{2}{5}\].

Step 4: Simplify the Expression

First, address the whole number: \( -2 \approx \frac{-20}{10} \). Now, compute a common denominator of 20 for all fractions: - \( \frac{-1}{2} = \frac{-10}{20}\)- \( \frac{1}{4} = \frac{5}{20}\)- \( \frac{2}{5} = \frac{8}{20}\)Now substitute these into the expression:\[ \frac{-20}{10} + \frac{-10}{20} + 0 + \frac{5}{20} + \frac{8}{20} = \frac{-40}{20} + \frac{-10+5+8}{20} = \frac{-40}{20} + \frac{3}{20} = \frac{-37}{20} \].

Step 5: Evaluate the Result

The evaluated sum of the terms is \(\frac{-37}{20}\). Ensure that the answer simplifies or recalculates any inaccuracies, though this fraction cannot be simplified further.

Key Concepts

Evaluating Expressions

Evaluating expressions involves finding the value of an algebraic expression by substituting numerical values for the given variables. In the context of summation in algebra, it means computing the result of a series of terms by inserting these specific values.

Consider the expression \( \sum_{i=1}^{5} \frac{x_i}{x_i+3} \). Here, you need to substitute the given set of \( x \) values into the expression. Let's break this down:

• Identify each \( x_i \) from the list: \( x_1 = -2, x_2 = -1, x_3 = 0, x_4 = 1, \) and \( x_5 = 2 \).
• Substitute each \( x_i \) into \( \frac{x_i}{x_i+3} \) to obtain individual values.

By evaluating, we mean computing these individual parts step by step to ensure the process is both precise and clear. This systematic approach helps solve even the most complex expressions.

Fractions in Mathematics

Fractions in mathematics represent parts of a whole and can be complex to manage, especially when they involve a summation. Here, each term \( \frac{x_i}{x_i+3} \) results in a fraction which must be evaluated precisely for the entire sum.

• The fraction \( \frac{-2}{1} = -2 \) seems simple, but others like \( \frac{-1}{2} \) require multiplication to find a common denominator for addition later.
• Converting each fraction to a common denominator simplifies the eventual summation. For example, \( \frac{-1}{2} \), \( \frac{1}{4} \), and \( \frac{2}{5} \) were all converted to have a common denominator of 20.

Understanding how to manipulate fractions by finding common denominators is crucial. It facilitates the addition process by creating equivalent fractions that can be easily added together.

Step-by-Step Problem Solving

Step-by-step problem solving embraces a structured approach to tackling problems, focusing on breaking down tasks into smaller, manageable parts. In mathematics, this method is especially beneficial when dealing with complex calculations, such as summation.

• Begin by identifying each component or term in the expression that needs evaluation.
• Calculate the value of each term independently before proceeding to aggregate their values.

For instance, solving the expression \( \sum_{i=1}^{5} \frac{x_i}{x_i+3} \) involved substituting each value of \( x_i \), evaluating them, and then adding their results.

This meticulous, methodical approach serves not only the current challenge but builds skills relevant for future mathematical endeavors. By working step-by-step, you gain confidence in handling intricate problems, ensuring all parts are correctly evaluated and summed.