Problem 12 - College Algebra with Modeling and Visualization

Question

Find the first four terms of the sequence. \(a_{n}=\frac{1}{n}+\frac{1}{3 n}\)

Step-by-Step Solution

Answer

The first four terms are \(\frac{4}{3}, \frac{2}{3}, \frac{4}{9}, \frac{1}{3}\).

Step 1: Find the first term

To find the first term of the sequence, substitute \(n = 1\) into the formula: \[a_1 = \frac{1}{1} + \frac{1}{3 \times 1} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}\] Thus, the first term \(a_1\) is \(\frac{4}{3}\).

Step 2: Find the second term

Next, substitute \(n = 2\) into the formula to calculate the second term:\[a_2 = \frac{1}{2} + \frac{1}{3 \times 2} = \frac{1}{2} + \frac{1}{6}\]To add these fractions, find a common denominator, which is 6: \[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{6} = \frac{1}{6} \]Thus,\[ a_2 = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\]The second term \(a_2\) is \(\frac{2}{3}\).

Step 3: Find the third term

Substitute \(n = 3\) into the formula to find the third term:\[a_3 = \frac{1}{3} + \frac{1}{3 \times 3} = \frac{1}{3} + \frac{1}{9}\]Find a common denominator, which is 9:\[ \frac{1}{3} = \frac{3}{9}, \quad \frac{1}{9} = \frac{1}{9} \]Therefore,\[ a_3 = \frac{3}{9} + \frac{1}{9} = \frac{4}{9}\]The third term \(a_3\) is \(\frac{4}{9}\).

Step 4: Find the fourth term

Substitute \(n = 4\) into the sequence formula for the fourth term:\[a_4 = \frac{1}{4} + \frac{1}{3 \times 4} = \frac{1}{4} + \frac{1}{12}\]The common denominator is 12:\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{12} = \frac{1}{12} \]Thus,\[ a_4 = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}\]Therefore, the fourth term \(a_4\) is \(\frac{1}{3}\).

Key Concepts

Fraction Addition

Adding fractions is a fundamental concept in arithmetic, essential for manipulating expressions and solving equations that involve fractions. When we want to add fractions like \( \frac{1}{2} \) and \( \frac{1}{3} \), the process isn't as simple as adding whole numbers, due to different denominators in the fractions. Here are some steps to follow:

• Identify the denominators of each fraction. Denominators are numbers located below the line in a fraction.
• Find a common denominator, which is a shared multiple of the two denominators.
• Convert each fraction so they both have this common denominator.
• Add the numerators (the top numbers) of the converted fractions while keeping the common denominator the same.
• Simplify the result, if possible, to make it easier to understand and work with.

For example, when dealing with \( \frac{1}{2} + \frac{1}{3} \), the common denominator is 6. Convert each fraction: \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \), then add: \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \).

Common Denominator

A common denominator is crucial when adding or subtracting fractions, as it provides a shared basis to combine different fractional expressions.To find a common denominator:

• Determine the least common multiple (LCM) of the denominators you are dealing with.
• Adjust each fraction so that it is equivalent to another fraction with the common denominator.
• Multiply the numerator and the denominator of the fraction by the same number to keep the value of the fraction unchanged.

Using the exercise example, to add \( \frac{1}{3} \) and \( \frac{1}{9} \), we identify 9 as the common denominator. Thus, \( \frac{1}{3} \rightarrow \frac{3}{9} \) and \( \frac{1}{9} \) remains as it is. Now the fractions can be easily added: \( \frac{3}{9} + \frac{1}{9} = \frac{4}{9} \). This ensures the arithmetic operations on fractions remain consistent and accurate.

Sequence Terms

In mathematics, a sequence is an ordered list of numbers that typically follow a specific pattern or rule. Each number in the sequence is referred to as a term. Understanding sequence terms is essential for recognizing numerical patterns that arise in various mathematical contexts.Here's how to find sequence terms for a particular sequence:

• Identify the sequence rule or formula that defines the sequence.
• Substitute the position of the term (like 1st, 2nd, 3rd, etc.) into the formula.
• Calculate the value of the sequence term by following the operations in the formula.

For the given exercise, the sequence formula is \( a_n = \frac{1}{n} + \frac{1}{3n} \). By substituting values such as \( n = 1 \), \( n = 2 \), etc., into the formula, we can find the terms of the sequence. For example, substituting \( n = 1 \), gives \( a_1 = \frac{4}{3} \), signifying the first term. This approach helps uncover the logic behind sequence creation and application.