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Based on the pattern, what are the next two terms of the sequence?


The given sequence is:

\[
9, \frac{9}{5}, \frac{9}{25}, \frac{9}{125}, \frac{9}{625}, \dots
\]

We can observe a clear pattern in the denominators:

- The first term has no denominator (or can be seen as \( \frac{9}{1} \)).
- The second term has \( 5 \) in the denominator.
- The third term has \( 25 = 5^2 \) in the denominator.
- The fourth term has \( 125 = 5^3 \) in the denominator.
- The fifth term has \( 625 = 5^4 \) in the denominator.

It seems that the denominator is increasing as powers of 5. The next two terms should have \( 5^5 = 3125 \) and \( 5^6 = 15625 \) in the denominator.

Thus, the next two terms of the sequence are:

\[
\frac{9}{3125}, \quad \frac{9}{15625}
\]

Now, let's check the options:

- Option a: \( \frac{9}{3125}, \frac{9}{3130} \)
- Option b: \( \frac{9}{630}, \frac{9}{3130} \)
- Option c: \( \frac{9}{3125}, \frac{9}{15625} \)
- Option d: \( \frac{9}{630}, \frac{9}{635} \)

The correct answer is **option c**.

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you identify patterns in numerical sequences?
2. What are arithmetic and geometric progressions?
3. How are powers and exponents used in sequences?
4. Can sequences also be used to model real-world scenarios?
5. What is the formula for the general term of a geometric sequence?

**Tip**: When identifying patterns in sequences, always check for operations (addition, multiplication, etc.) applied consistently to terms.

Based on the pattern, what are the next two terms of the sequence?

Understand how to find the next terms in a geometric sequence where the denominators are powers of 5. This problem uses basic sequence patterns to predict the following terms. Ideal for students in Grades 6-8.

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