We are given a sequence: 18, 6, 2, $\frac{2}{3}$, $\frac{2}{9}$, and so on. This appears to be a geometric sequence because the ratio between consecutive terms is constant.

Step 1: Finding the common ratio

• The first term is $18$.
• The second term is $6$, so the ratio between the first and second terms is: $\frac{6}{18} = \frac{1}{3}.$
• The third term is $2$, so the ratio between the second and third terms is: $\frac{2}{6} = \frac{1}{3}.$

Thus, the common ratio $r$ of this geometric sequence is $\frac{1}{3}$.

Step 2: Formula for the general term of a geometric sequence

The general term $S(n)$ of a geometric sequence is given by: $S(n) = a \cdot r^{n-1},$ where:

• $a$ is the first term,
• $r$ is the common ratio, and
• $n$ is the position of the term.

In this case:

• $a = 18$ (the first term),
• $r = \frac{1}{3}$ (the common ratio).

Thus, the formula for this sequence is: $S(n) = 18 \left( \frac{1}{3} \right)^{n-1}.$

Step 3: Matching with the options

• The correct function is represented by option D: $S(n) = 18 \left( \frac{1}{3} \right)^{n-1}.$

Would you like further details or explanations? Here are five related questions for more practice:

1. How do you identify whether a sequence is arithmetic or geometric?
2. What is the sum formula for a finite geometric series?
3. What happens if the common ratio $r > 1$ in a geometric sequence?
4. How would you find the sum of an infinite geometric series?
5. How can you find the 10th term of a geometric sequence given the first term and common ratio?

Tip: In a geometric sequence, the ratio between consecutive terms remains constant, which makes it easy to identify and derive the general term formula.