To solve this problem, let's analyze the given geometric sequence:

Sequence: $27, 9, 3, 1, \dots$

1. Step 1: Identify the common ratio ($r$): To find the common ratio, divide any term by its preceding term: $r = \frac{9}{27} = \frac{1}{3}$ (Similarly, $\frac{3}{9} = \frac{1}{3}$ confirms the consistency of the ratio.)

2. Step 2: Write the general term formula: The formula for the $n$-th term ($a_n$) of a geometric sequence is: $a_n = a_1 \cdot r^{n-1}$ Here:

   • $a_1 = 27$ (the first term),
   • $r = \frac{1}{3}$.

   Substituting these values, the formula becomes: $a_n = 27 \cdot \left(\frac{1}{3}\right)^{n-1}$

3. Step 3: Find $a_7$: Substitute $n = 7$ into the formula: $a_7 = 27 \cdot \left(\frac{1}{3}\right)^{7-1} = 27 \cdot \left(\frac{1}{3}\right)^6$ Simplify: $\left(\frac{1}{3}\right)^6 = \frac{1}{3^6} = \frac{1}{729}$ Thus: $a_7 = 27 \cdot \frac{1}{729} = \frac{27}{729} = \frac{1}{27}$

Final Answers:

• General term: $a_n = 27 \cdot \left(\frac{1}{3}\right)^{n-1}$
• Seventh term: $a_7 = \frac{1}{27}$

Would you like further clarification or additional examples?
Here are five related questions to explore:

1. How do we determine if a sequence is geometric versus arithmetic?
2. Can the common ratio be negative in a geometric sequence? What would that look like?
3. How would the formula change if the sequence started with a negative first term?
4. What happens to the terms as $n$ approaches infinity in this sequence?
5. How can this formula be adapted for a geometric sequence with a fractional starting term?

Tip: For geometric sequences, always verify the common ratio by dividing consecutive terms!