Find the sum of the geometric sequence where the first term is 3, the last term is 46 875, and the common ratio is 5.

To find the sum of a geometric sequence, we can use the formula:

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

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

In this case:
- The first term \(a = 3\),
- The last term \(l = 46,875\),
- The common ratio \(r = 5\).

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

\[
l = a \cdot r^{n-1}
\]

Substituting the known values:

\[
46,875 = 3 \cdot 5^{n-1}
\]

Let's solve for \(n\).The number of terms \(n\) in the sequence is 7.

Now, we can use the sum formula for the geometric sequence:

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

Substituting the known values (\(a = 3\), \(r = 5\), and \(n = 7\)):

\[
S_7 = \frac{3(1 - 5^7)}{1 - 5}
\]

Let me calculate this.The sum of the geometric sequence is \(58,593\).

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the common ratio in a geometric sequence?
2. What is the sum formula for an infinite geometric series?
3. How would the sum change if the first term doubled?
4. How can you find the number of terms if the sum is given instead of the last term?
5. What happens to the sum if the common ratio is less than 1?

**Tip:** In geometric sequences, the ratio between consecutive terms remains constant, which makes it easy to predict and calculate large sequences.

Learn how to find the sum of a geometric sequence with a first term of 3, a last term of 46,875, and a common ratio of 5. This problem explores key algebraic concepts and provides a detailed solution using geometric sequence formulas. Suitable for Grades 9-12.

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