Problem 19 - Algebra for College Students

Question

\text { The 8th term of } \frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, \ldots, \frac{1}{4374}

Step-by-Step Solution

Answer

The 8th term is \( \frac{1}{4374} \).

Step 1: Identify the Sequence

First, observe that the sequence provided is a shrinking sequence of fractions: \( \frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, \ldots \). These fractions suggest a geometric sequence where each term after the first is multiplied by a common ratio.

Step 2: Determine the Common Ratio

To find the common ratio, divide the second term by the first term: \( \frac{1/6}{1/2} = \frac{1}{6} \times \frac{2}{1} = \frac{2}{6} = \frac{1}{3} \). Similarly, dividing the third term by the second, \( \frac{1/18}{1/6} = \frac{6}{18} = \frac{1}{3} \). Thus, the common ratio \( r \) is \( \frac{1}{3} \).

Step 3: Set Up the General Term Formula

For a geometric sequence, the general term \( a_n \) can be defined by the formula \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step 4: Calculate the 8th Term

Substitute the values of the first term \( a_1 = \frac{1}{2} \), the common ratio \( r = \frac{1}{3} \), and \( n = 8 \) into the general formula to find the 8th term: \[ a_8 = \frac{1}{2} \cdot \left(\frac{1}{3}\right)^{7} \]Calculate the power: \[ \left(\frac{1}{3}\right)^{7} = \frac{1}{2187} \]Now, find the 8th term by substitution:\[ a_8 = \frac{1}{2} \cdot \frac{1}{2187} = \frac{1}{4374} \]

Step 5: Verify the Result

As the result \( \frac{1}{4374} \) is consistent with the sequence pattern and the geometric sequence calculations, we can conclude the answer is correct.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a key component. It is the factor by which each term in the sequence is multiplied to obtain the next term. For example, in the sequence \( \frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, \ldots \), we can determine the common ratio by dividing any term by its preceding term. This helps maintain the uniformity of the sequence.

• Take the second term \( \frac{1}{6} \) and divide it by the first term \( \frac{1}{2} \).
• Perform the division: \( \frac{1/6}{1/2} = \frac{1}{6} \times \frac{2}{1} = \frac{1}{3} \).

Once the common ratio \( r \) is known, which in our case is \( \frac{1}{3} \), we can easily calculate subsequent terms by applying this ratio.

General Term Formula in Geometric Sequences

To find any term in a geometric sequence, we can use the general term formula. This formula is a powerful tool for calculating not just the next term, but any term within the sequence quickly and efficiently. The formula is given by: \[ a_n = a_1 \cdot r^{(n-1)} \]where:

• \( a_n \) is the term you wish to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the term number.

Using this formula ensures you can directly jump to the term number you are interested in without having to calculate all prior terms.

Calculating Specific Terms in a Geometric Sequence

Suppose we want to find the 8th term of our sequence \( \frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, \ldots \). By using the general term formula \( a_n = a_1 \cdot r^{(n-1)} \), we can easily calculate specific terms. Let's go through the steps for the 8th term:1. Identify the known values:

• First term \( a_1 = \frac{1}{2} \)
• Common ratio \( r = \frac{1}{3} \)
• Desired term number \( n = 8 \)

2. Replace these values into the formula: \[ a_8 = \frac{1}{2} \cdot \left(\frac{1}{3}\right)^{7} \]3. Calculate the power: \[ \left(\frac{1}{3}\right)^{7} = \frac{1}{2187} \]4. Substitute back: \[ a_8 = \frac{1}{2} \cdot \frac{1}{2187} = \frac{1}{4374} \]Therefore, the 8th term of the sequence is \( \frac{1}{4374} \), demonstrating how geometric sequence calculations work seamlessly with the formula.