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.