91Ó°ÊÓ

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.

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.

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.