https://file.aiquickdraw.com/mathbot/file/1734136132161lo2brcmr.jpg

To solve the problem, let's break it down step by step.

### Problem:
We are given:
1. The common ratio \( r = \frac{1}{6} \).
2. The third term \( a_3 = 12 \).
3. We need to find the **first term** (\( a_1 \)) and the **seventh term** (\( a_7 \)).

### General Formula for a Geometric Sequence:
The \( n \)-th term of a geometric sequence is given by:
\[
a_n = a_1 \cdot r^{n-1}
\]

#### Step 1: Use the third term to find \( a_1 \).
For \( n = 3 \), the formula becomes:
\[
a_3 = a_1 \cdot r^{3-1} = a_1 \cdot r^2
\]
Substitute \( a_3 = 12 \) and \( r = \frac{1}{6} \):
\[
12 = a_1 \cdot \left(\frac{1}{6}\right)^2
\]
Simplify \( \left(\frac{1}{6}\right)^2 \):
\[
12 = a_1 \cdot \frac{1}{36}
\]
Solve for \( a_1 \):
\[
a_1 = 12 \cdot 36 = 432
\]

#### Step 2: Find the seventh term (\( a_7 \)).
For \( n = 7 \), the formula becomes:
\[
a_7 = a_1 \cdot r^{7-1} = a_1 \cdot r^6
\]
Substitute \( a_1 = 432 \) and \( r = \frac{1}{6} \):
\[
a_7 = 432 \cdot \left(\frac{1}{6}\right)^6
\]
Simplify \( \left(\frac{1}{6}\right)^6 \):
\[
\left(\frac{1}{6}\right)^6 = \frac{1}{6^6} = \frac{1}{46656}
\]
So:
\[
a_7 = 432 \cdot \frac{1}{46656} = \frac{432}{46656}
\]
Simplify the fraction:
\[
a_7 = \frac{1}{108}
\]

### Final Answer:
- **First term (\( a_1 \))**: \( 432 \)
- **Seventh term (\( a_7 \))**: \( \frac{1}{108} \)

Would you like a detailed explanation for any step? Let me know!

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### Related Questions:
1. How do you derive the general formula for a geometric sequence?
2. What happens to the terms if the common ratio is greater than 1?
3. How do you find the sum of the first \( n \) terms of a geometric sequence?
4. Can a geometric sequence have a common ratio of 0? Why or why not?
5. How does the position of a term affect its value in a geometric sequence?

### Tip:
When working with fractions in geometric sequences, simplify intermediate steps to avoid errors and make calculations easier!

The common ratio is 1/6, and the third term is 12. Find the first and seventh terms of the geometric sequence.

Solve a geometric sequence problem where the common ratio is 1/6, and the third term is 12. Learn step-by-step how to calculate the first term (432) and the seventh term (1/108) using the formula for geometric sequences.

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