Let's go through each question step-by-step to find the correct answers:

1. What expression is the same as $(a \cdot a \cdot a \cdot a)(b \cdot b)$?

Given: $(a \cdot a \cdot a \cdot a)(b \cdot b)$

Simplifying: $(a \cdot a \cdot a \cdot a) = a^4, \quad (b \cdot b) = b^2$ Therefore, the expression is: $a^4 b^2$

Answer: C. $a^4 b^2$

2. What is the common ratio of the sequence $36, 12, 4, \ldots$?

To find the common ratio $r$ of a geometric sequence, divide the second term by the first term: $r = \frac{12}{36} = \frac{1}{3}$

Answer: C. $\frac{1}{3}$

3. The first term of a geometric sequence is 5, and the second term is 10. What is the sixth term?

Given a geometric sequence where the first term $a_1 = 5$ and the second term $a_2 = 10$, we can find the common ratio $r$: $r = \frac{a_2}{a_1} = \frac{10}{5} = 2$

The $n$-th term of a geometric sequence is given by: $a_n = a_1 \cdot r^{n-1}$

For the sixth term ($n = 6$): $a_6 = 5 \cdot 2^{6-1} = 5 \cdot 2^5 = 5 \cdot 32 = 160$

Answer: A. 160

4. The common ratio of a geometric sequence is $\frac{1}{2}$, and the fourth term is $\frac{1}{4}$. What is the third term of the sequence?

Let the third term be $a_3$. Since the common ratio $r = \frac{1}{2}$, the fourth term can be expressed as: $a_4 = a_3 \cdot \frac{1}{2} = \frac{1}{4}$

Solving for $a_3$: $a_3 = \frac{1}{4} \cdot 2 = \frac{1}{2}$

Answer: B. $\frac{1}{2}$

5. The common ratio of a geometric sequence is $\frac{3}{2}$. Find the first three terms if the fifth term is 1.

Let the first term be $a_1 = a$. The $n$-th term of a geometric sequence is given by: $a_n = a_1 \cdot r^{n-1}$

For the fifth term: $a_5 = a \cdot \left(\frac{3}{2}\right)^4 = 1$

Solving for $a$: $a \cdot \frac{81}{16} = 1 \implies a = \frac{16}{81}$

Now, the first three terms are: $a_1 = \frac{16}{81}, \quad a_2 = \frac{16}{81} \cdot \frac{3}{2} = \frac{8}{27}, \quad a_3 = \frac{8}{27} \cdot \frac{3}{2} = \frac{4}{9}$

Answer: A. $\frac{16}{81}, \frac{8}{27}, \frac{4}{9}$

Final Answers:

1. C. $a^4 b^2$
2. C. $\frac{1}{3}$
3. A. 160
4. B. $\frac{1}{2}$
5. A. $\frac{16}{81}, \frac{8}{27}, \frac{4}{9}$

Would you like more details on any of the solutions or have any questions?

Related Questions:

1. How do you derive the general formula for the $n$-th term of a geometric sequence?
2. What are the properties of a geometric sequence with a negative common ratio?
3. How do you find the sum of a finite geometric series?
4. How do you convert a repeating decimal into a fraction using geometric series?
5. What are the conditions for convergence of an infinite geometric series?

Tip: For geometric sequences, always check the sign and magnitude of the common ratio to determine the nature of the sequence (increasing, decreasing, or alternating).