Write an explicit formula for a, start subscript, n, end subscripta
n
​
, the n, start superscript, th, end superscriptn
th
term of the sequence 18, comma, 6, comma, 2, comma, point, point, .18,6,2,....

The given sequence is: 18, 6, 2, ...

To find an explicit formula for the \(n\)th term of this sequence, let's first identify the pattern.

### Step 1: Identify the Pattern
- The first term \(a_1 = 18\).
- The second term \(a_2 = 6\).
- The third term \(a_3 = 2\).

The sequence seems to be decreasing by a factor each time. Let's check:

\[
\text{Ratio from } a_1 \text{ to } a_2 = \frac{6}{18} = \frac{1}{3}
\]
\[
\text{Ratio from } a_2 \text{ to } a_3 = \frac{2}{6} = \frac{1}{3}
\]

So, this is a geometric sequence where each term is multiplied by \(\frac{1}{3}\) to get the next term.

### Step 2: Write the General Formula
For a geometric sequence, the \(n\)th term \(a_n\) can be given by:

\[
a_n = a_1 \times r^{(n-1)}
\]

where \(a_1\) is the first term and \(r\) is the common ratio.

Here:
- \(a_1 = 18\)
- \(r = \frac{1}{3}\)

Thus, the explicit formula for the \(n\)th term is:

\[
a_n = 18 \times \left(\frac{1}{3}\right)^{n-1}
\]

This formula will give you the \(n\)th term of the sequence.

Would you like to explore this further or have any questions?

Here are 5 related questions:
1. What is the 5th term of this sequence?
2. How would the sequence change if the common ratio were \(\frac{1}{2}\) instead of \(\frac{1}{3}\)?
3. What is the sum of the first 10 terms of this sequence?
4. Can you derive a general formula for the sum of the first \(n\) terms of a geometric sequence?
5. How does the value of \(r\) affect the convergence of an infinite geometric series?

**Tip:** In a geometric sequence, if the common ratio \(r\) is between -1 and 1 (excluding 0), the sequence will converge to 0 as \(n\) increases.

Learn how to find the nth term of the geometric sequence 18, 6, 2, ... using the explicit formula. This problem covers key concepts in geometric sequences, including identifying the common ratio and using the nth term formula. Suitable for students in Grades 10-12.

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