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The sequence given is:

\[ 25, 75, 225, 675, \dots \]

This is a geometric sequence because each term is being multiplied by a constant factor to get the next term. To find the common ratio (\(r\)), we divide the second term by the first:

\[
r = \frac{75}{25} = 3
\]

Thus, the common ratio is \(3\).

To find the 6th term, we use the general formula for the \(n\)-th term of a geometric sequence:

\[
a_n = a_1 \times r^{(n-1)}
\]

Where:
- \(a_1 = 25\) (the first term),
- \(r = 3\) (the common ratio),
- \(n = 6\).

Now, let's calculate the 6th term:

\[
a_6 = 25 \times 3^{(6-1)} = 25 \times 3^5 = 25 \times 243 = 6075
\]

The 6th term is **6075**.

Would you like further details or have any questions? Here are five related questions you might find useful:

1. How do you find the sum of the first \(n\) terms of a geometric sequence?
2. How can you determine if a sequence is arithmetic or geometric?
3. What is the difference between the formula for arithmetic and geometric sequences?
4. Can the common ratio in a geometric sequence be negative, and what would that look like?
5. How would you calculate the 10th term in this sequence?

**Tip:** For geometric sequences, the common ratio plays a crucial role in determining how rapidly the sequence grows or shrinks. Always verify the ratio by dividing consecutive terms.

Learn how to find the 6th term of the geometric sequence 25, 75, 225, 675, … with a detailed solution. This problem involves calculating the n-th term using the geometric sequence formula and is suitable for students in Grades 9-12.

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