Find the 7th term of the geometric sequence 
9, 3, 1,   .

To find the 7th term of a geometric sequence, we use the formula for the \(n\)th term of a geometric sequence:

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

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

From the sequence given (9, 3, 1, ...):
- The first term \(a_1 = 9\),
- The common ratio \(r = \frac{3}{9} = \frac{1}{3}\).

Now, using the formula to find the 7th term (\(n = 7\)):

\[
a_7 = 9 \cdot \left(\frac{1}{3}\right)^{7-1} = 9 \cdot \left(\frac{1}{3}\right)^6
\]

\[
a_7 = 9 \cdot \frac{1}{729} = \frac{9}{729} = \frac{1}{81}
\]

Thus, the 7th term of the geometric sequence is \(\frac{1}{81}\).

Would you like further details or have any questions?

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Here are 5 related questions to explore further:
1. How do you find the sum of the first \(n\) terms of a geometric sequence?
2. What happens if the common ratio is negative in a geometric sequence?
3. How do you determine if a sequence is arithmetic or geometric?
4. How do you find the general formula for any term in a geometric sequence?
5. What is the difference between finite and infinite geometric series?

**Tip:** Always double-check the common ratio before applying the geometric sequence formula.

Find the 7th term of the geometric sequence 9, 3, 1, ...

Discover how to find the 7th term of the geometric sequence starting with 9, 3, 1 using the geometric sequence formula. This detailed solution explores key concepts such as common ratios and exponents, suitable for students in Grades 8-10.

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