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.