II. Directions. Find the specified term of the geometric sequences given the first term
and the common ratio.
Given

It seems like your question was cut off at the end. However, I'll explain the process of finding a specific term in a geometric sequence when the first term (\(a_1\)) and the common ratio (\(r\)) are given. If you provide the complete details, I can work on specific examples.

### General Formula for the \(n\)-th Term of a Geometric Sequence

Given:
- The first term \( a_1 \).
- The common ratio \( r \).
- The position of the term \( n \).

The formula to find the \(n\)-th term (\( a_n \)) of a geometric sequence is:

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

### Example Problems

1. **Find the 5th term when \( a_1 = 3 \) and \( r = 2 \)**:

   Using the formula:

   \[
   a_5 = 3 \cdot 2^{(5-1)} = 3 \cdot 2^4 = 3 \cdot 16 = 48
   \]

2. **Find the 8th term when \( a_1 = 5 \) and \( r = -3 \)**:

   Using the formula:

   \[
   a_8 = 5 \cdot (-3)^{(8-1)} = 5 \cdot (-3)^7 = 5 \cdot (-2187) = -10935
   \]

3. **Find the 4th term when \( a_1 = 10 \) and \( r = 0.5 \)**:

   Using the formula:

   \[
   a_4 = 10 \cdot (0.5)^{(4-1)} = 10 \cdot (0.5)^3 = 10 \cdot 0.125 = 1.25
   \]

4. **Find the 6th term when \( a_1 = -2 \) and \( r = 4 \)**:

   Using the formula:

   \[
   a_6 = -2 \cdot 4^{(6-1)} = -2 \cdot 4^5 = -2 \cdot 1024 = -2048
   \]

5. **Find the 3rd term when \( a_1 = 12 \) and \( r = \frac{1}{3} \)**:

   Using the formula:

   \[
   a_3 = 12 \cdot \left(\frac{1}{3}\right)^{(3-1)} = 12 \cdot \left(\frac{1}{3}\right)^2 = 12 \cdot \frac{1}{9} = \frac{12}{9} = \frac{4}{3}
   \]

Let me know if you have specific sequences you need help with, or if you'd like more detailed examples! 

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Would you like details or have any questions? Here are some related questions to consider:

1. How do you determine the common ratio when given two consecutive terms?
2. What happens when the common ratio is a fraction or a negative number?
3. How can you find the sum of a finite geometric series?
4. How does the formula change if the sequence is infinite?
5. How do you identify if a sequence is geometric given a set of terms?

**Tip:** Always double-check your calculations, especially when dealing with exponents and negative ratios, as small mistakes can lead to large errors in the results.

Learn how to find a specific term in geometric sequences using the formula \( a_n = a_1 \cdot r^{(n-1)} \). Includes detailed examples with different values of the first term and common ratio. Suitable for high school students studying geometric sequences.

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