Problem 53 - 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 \(x_{5}=2\) $$\sum_{i=2}^{5} \frac{x_{i}+1}{x_{i}+2}$$

Step-by-Step Solution

Answer

The sum is \(\frac{23}{12}\).

Step 1: Identify Terms of the Sum

The given sum is \(\sum_{i=2}^{5} \frac{x_{i}+1}{x_{i}+2}\). We need to evaluate this sum for each term where \(i\) ranges from 2 to 5. The terms we will evaluate are for \(x_{2}, x_{3}, x_{4},\) and \(x_{5}\).

Step 2: Calculate the First Term

For \(i=2\), \(x_{2} = -1\). Calculate the first term: \[ \frac{-1+1}{-1+2} = \frac{0}{1} = 0. \]

Step 3: Calculate the Second Term

For \(i=3\), \(x_{3} = 0\). Calculate the second term: \[ \frac{0+1}{0+2} = \frac{1}{2}. \]

Step 4: Calculate the Third Term

For \(i=4\), \(x_{4} = 1\). Calculate the third term: \[ \frac{1+1}{1+2} = \frac{2}{3}. \]

Step 5: Calculate the Fourth Term

For \(i=5\), \(x_{5} = 2\). Calculate the fourth term: \[ \frac{2+1}{2+2} = \frac{3}{4}. \]

Step 6: Sum the Calculated Terms

Add all the evaluated terms: \[ 0 + \frac{1}{2} + \frac{2}{3} + \frac{3}{4}. \]

Step 7: Find the Common Denominator and Sum

The common denominator for the fractions \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}\) is 12. Convert and sum them: \[ \frac{1}{2} = \frac{6}{12}, \quad \frac{2}{3} = \frac{8}{12}, \quad \frac{3}{4} = \frac{9}{12}. \]\[ \frac{6}{12} + \frac{8}{12} + \frac{9}{12} = \frac{23}{12}. \]

Step 8: Combine and Simplify the Sum

Add all terms: \[ 0 + \frac{23}{12} = \frac{23}{12}. \] The final result is \(\frac{23}{12}\).

Key Concepts

Arithmetic Series

An arithmetic series is a sum of terms in a sequence, where each term after the first is generated by adding a fixed number, known as the common difference, to the previous term. In our exercise, although the sum given is not a typical arithmetic series due to the operations involved, understanding the arithmetic progression can provide insight into series summation.

• An arithmetic series is represented as \( a, a+d, a+2d, a+3d, \ldots \)
• The sum formula for an arithmetic series is \( S_n = \frac{n}{2} (2a + (n-1)d) \)
• This doesn't apply directly to fractional sums, but arithmetic understanding is vital for manipulating series.

In more complex expressions like our sum, recognizing patterns or consistent intervals introduces the notion of logical progression, which can streamline tackling similar problems.

Fraction Operations

Working with fractions involves various operations including addition, subtraction, multiplication, and division. In this exercise, we were required to perform addition on fractions which arose from each calculated term.

• Always simplify fractions wherever possible.
• To add fractions, they must have the same denominator.

It's useful to remember basic operations:

• For addition: \( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \)
• Simplification should be done when common factors exist in the numerator and the denominator.

Fraction operations appear frequently in summation problems, making their clear understanding crucial.

Common Denominator

Identifying a common denominator is a key step in adding fractions. A common denominator is a shared multiple of the denominators you are working with. For the fractions in our exercise: \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \), determining the common denominator facilitates the addition process.

• Find the least common multiple (LCM) for all denominators involved.
• Convert each fraction to an equivalent with this LCM as the new denominator.

The LCM for the denominators 2, 3, and 4 is 12. Thus:

• \( \frac{1}{2} = \frac{6}{12} \)
• \( \frac{2}{3} = \frac{8}{12} \)
• \( \frac{3}{4} = \frac{9}{12} \)

This process ensures an accurate sum of the fractions, enabling them to be added seamlessly.