Find the sum of the terms of a geometric sequence where the first term is 4, the last term is 324, and the common ratio is 3.

The sum of the terms of a geometric sequence can be found using the formula:

\[
S_n = \frac{a_1 (1 - r^n)}{1 - r}
\]

where:
- \( S_n \) is the sum of the terms,
- \( a_1 \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

In this problem:
- \( a_1 = 4 \),
- \( a_n = 324 \) (the last term),
- \( r = 3 \).

First, we need to find \( n \), the number of terms in the sequence. The general formula for the \(n\)-th term of a geometric sequence is:

\[
a_n = a_1 \cdot r^{n-1}
\]

Substitute the known values:

\[
324 = 4 \cdot 3^{n-1}
\]

Now, solve for \( n \):

\[
324 = 4 \cdot 3^{n-1} \implies 81 = 3^{n-1}
\]

Since \( 81 = 3^4 \), we have:

\[
3^{n-1} = 3^4 \implies n-1 = 4 \implies n = 5
\]

Now that we know \( n = 5 \), we can calculate the sum of the first 5 terms using the sum formula:

\[
S_5 = \frac{4(1 - 3^5)}{1 - 3}
\]

Simplify:

\[
S_5 = \frac{4(1 - 243)}{-2} = \frac{4(-242)}{-2} = \frac{-968}{-2} = 484
\]

Thus, the sum of the terms of the geometric sequence is \( 484 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the sum of an infinite geometric series if \( |r| < 1 \)?
2. How do you find the number of terms in an arithmetic sequence?
3. What is the general formula for the sum of an arithmetic sequence?
4. How does the formula for the sum of a geometric sequence change if the ratio \( r \) is between -1 and 1?
5. What are the differences between arithmetic and geometric sequences?

**Tip**: Always check whether the ratio \( r \) is greater than 1, less than 1, or negative to determine the behavior of the sequence.

Learn how to calculate the sum of a geometric sequence where the first term is 4, the last term is 324, and the common ratio is 3. This problem includes key concepts like the formula for the sum of a geometric sequence and finding the number of terms. Suitable for students in Grades 9-10.

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