To solve the problem of finding the 13th term of the given geometric sequence:

The sequence is: $2, 6, 18, 54, \ldots$

Step 1: Identify the common ratio ($r$)

To find the common ratio ($r$) of the geometric sequence, divide the second term by the first term: $r = \frac{6}{2} = 3$ This means that each term is multiplied by 3 to get the next term.

Step 2: Use the formula for the $n$-th term of a geometric sequence

The formula for the $n$-th term ($a_n$) of a geometric sequence is: $a_n = a_1 \times r^{n-1}$ where:

• $a_1$ is the first term of the sequence,
• $r$ is the common ratio, and
• $n$ is the term number.

Step 3: Substitute values to find the 13th term

Here, $a_1 = 2$, $r = 3$, and $n = 13$. Substituting into the formula: $a_{13} = 2 \times 3^{12}$

Step 4: Calculate $3^{12}$

$3^{12} = 531,441$

Step 5: Multiply by 2

$a_{13} = 2 \times 531,441 = 1,062,882$

Final Answer:

The 13th term of the sequence is $1,062,882$.

Do you need more details or have any questions?

Here are 5 related questions to expand on this:

1. How would the formula change if the common ratio was negative?
2. What if you were asked to find the sum of the first 13 terms of the sequence?
3. How can geometric sequences be applied in real-world scenarios?
4. What happens to the geometric sequence if the common ratio is a fraction?
5. How would the general term formula change if the first term was negative?

Tip: When dealing with geometric sequences, always verify the common ratio by checking multiple terms to avoid errors.