Problem 32 - A Graphical Approach to College Algebra

Question

Find the sum for each series. $$\sum_{i=1}^{5} \frac{1}{i+1}$$

Step-by-Step Solution

Answer

The sum is \( \frac{29}{20} \) or 1\( \frac{9}{20} \).

Step 1: Understand the Series

The series is given by \( \sum_{i=1}^{5} \frac{1}{i+1} \). This means that we need to find the sum of the fractions \( \frac{1}{i+1} \) where \( i \) ranges from 1 to 5.

Step 2: List the Terms

Substitute \( i \) from 1 to 5 into the expression \( \frac{1}{i+1} \):- For \( i = 1 \), the term is \( \frac{1}{2} \).- For \( i = 2 \), the term is \( \frac{1}{3} \).- For \( i = 3 \), the term is \( \frac{1}{4} \).- For \( i = 4 \), the term is \( \frac{1}{5} \).- For \( i = 5 \), the term is \( \frac{1}{6} \).Thus, the terms to be summed are \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6} \).

Step 3: Calculate the Sum

Add up all the fractions: \[ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} \].Find a common denominator, which for these fractions is 60:- \( \frac{1}{2} = \frac{30}{60} \)- \( \frac{1}{3} = \frac{20}{60} \)- \( \frac{1}{4} = \frac{15}{60} \)- \( \frac{1}{5} = \frac{12}{60} \)- \( \frac{1}{6} = \frac{10}{60} \)Now, add these fractions: \( \frac{30}{60} + \frac{20}{60} + \frac{15}{60} + \frac{12}{60} + \frac{10}{60} \).This sums to \( \frac{87}{60} \).

Step 4: Simplify the Result

Simplify the fraction \( \frac{87}{60} \).Factor both the numerator and the denominator to determine the greatest common divisor (GCD).\( 87 = 3 \times 29 \)\( 60 = 2 \times 2 \times 3 \times 5 \).The GCD is 3. Divide the numerator and the denominator by 3:\( \frac{87}{60} = \frac{29}{20} \).

Step 5: Express as a Mixed Number (Optional)

Convert \( \frac{29}{20} \) to a mixed number.\( 29 \div 20 = 1 \) with a remainder of 9, so \( \frac{29}{20} = 1 \frac{9}{20} \).

Key Concepts

Series

A series can be thought of as the sum of the terms of a sequence. When you are given a series, your task is to add up all the terms specified in the sequence. Series can be finite or infinite, depending on how many terms you include. In the given example, the series is finite with five terms.

• The notation for a series often uses the Greek letter sigma (\( \Sigma \)), indicating summation.
• For example, \( \sum_{i=1}^{5} \frac{1}{i+1} \) means to find the sum of the terms as \( i \) runs from 1 to 5.

Understanding series is crucial, especially in mathematical fields where you want to find totals or aggregate amounts.
Always remember, the focus is on the addition of terms, making series vital for certain calculations and analyses in mathematics.

Fractions

Fractions represent parts of a whole. In a mathematical context, they show division of numerators (top numbers) by denominators (bottom numbers). Working with fractions requires an understanding of how to manipulate them, particularly in arithmetic operations like addition or subtraction.

• Before adding fractions, you need a common denominator. This involves finding a shared multiple which allows all fractions to be expressed equivalently.
• For instance, when summing \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \) and \( \frac{1}{6} \), the common denominator is 60.
• After converting the fractions to have this common denominator, add the numerators together to get the result.
• The last step is to simplify, which involves dividing both the numerator and denominator by their greatest common divisor.

Fractions are fundamental in mathematics as they allow precise representation and calculation of quantities that are not whole numbers.

Arithmetic Sequence

An arithmetic sequence is a type of sequence in mathematics where the difference between consecutive terms is constant. This constant difference is called the "common difference."

• If a sequence is arithmetic, it means each term is derived by adding a fixed, constant number to the previous term.
• The formula for finding the \( n \)-th term of an arithmetic sequence is given by: \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.
• However, in the given exercise, the sequence formed (\( \frac{1}{i+1} \)) is not arithmetic because the difference between consecutive terms is not constant.

Knowing how to identify whether a sequence is arithmetic can help in deciding the right approach and formula to use in various mathematical scenarios.