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!

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!