To find the sum of the terms of a geometric sequence, you can use the formula for the sum of the first $n$ terms of a geometric sequence:

$S_n = a_1 \frac{1 - r^n}{1 - r}$

where:

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

For the given sequence $6, 18, 54, 162, \ldots$:

• $a_1 = 6$
• $r = 3$

Sum of the First $n$ Terms

Using the formula, the sum of the first $n$ terms $S_n$ is:

$S_n = 6 \frac{1 - 3^n}{1 - 3}$

$S_n = 6 \frac{1 - 3^n}{-2}$

$S_n = -3 (1 - 3^n)$

$S_n = 3(3^n - 1)$

Example: Sum of the First 4 Terms

Let's calculate the sum of the first 4 terms:

$S_4 = 3(3^4 - 1)$

$S_4 = 3(81 - 1)$

$S_4 = 3 \times 80$

$S_4 = 240$

So, the sum of the first 4 terms is 240.

Would you like more details or have any questions?

Here are some related questions you might consider:

1. How can you find the sum of the first 10 terms of this sequence?
2. What is the sum of the first $n$ terms if the common ratio is a fraction?
3. How does the formula for the sum change if the common ratio $r$ is negative?
4. What is the sum to infinity for a geometric sequence, and when is it applicable?
5. Can the formula for the sum of a geometric series be derived from first principles?

Tip: For geometric sequences where $|r| < 1$, the sum to infinity can be calculated using $S = \frac{a_1}{1 - r}$, which is useful for series that converge.