This is a geometric sequence with the terms 2, 6, 18, 54, ..., and you are asked to find $a_{11}$.

Steps to find $a_{11}$:

1. Identify the first term ($a_1$) and the common ratio ($r$).

   • The first term $a_1 = 2$.
   • To find the common ratio $r$, divide the second term by the first term: $r = \frac{6}{2} = 3$

2. Use the formula for the $n$-th term of a geometric sequence: The general formula for the $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{(n-1)}$ Where:

   • $a_n$ is the $n$-th term.
   • $a_1$ is the first term.
   • $r$ is the common ratio.
   • $n$ is the term number.

3. Plug in the values for $a_{11}$: $a_{11} = 2 \cdot 3^{(11-1)} = 2 \cdot 3^{10}$

4. Calculate $3^{10}$: $3^{10} = 59049$

5. Now, calculate $a_{11}$: $a_{11} = 2 \cdot 59049 = 118098$

Thus, $a_{11} = 118098$.

Would you like further details on geometric sequences or have any questions? Here are some related questions:

1. How do you find the sum of a geometric sequence?
2. What happens if the common ratio is negative or between 0 and 1?
3. Can a geometric sequence have fractional terms?
4. How do you calculate the general formula for the sum of an infinite geometric series?
5. How do you find the common ratio of a geometric sequence when not directly given?

Tip: Always check if the sequence is truly geometric by ensuring the ratio between consecutive terms remains constant.