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The problem asks to determine the common ratio of each geometric sequence given the pairs of terms. Here is a detailed breakdown of the calculations:

### Problem: Determine the common ratio \( r \) of the geometric sequences given the pairs of terms below.

Let's solve each problem by using the formula for the \( n \)-th term of a geometric sequence:

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

where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, and \( r \) is the common ratio.

### 1. \( a_3 = 4 \) and \( a_5 = 21 \)
For this, use the formula for the terms:

\[
a_3 = a_1 \cdot r^2 = 4
\]
\[
a_5 = a_1 \cdot r^4 = 21
\]

Divide the second equation by the first:

\[
\frac{a_5}{a_3} = \frac{a_1 \cdot r^4}{a_1 \cdot r^2} = r^2 = \frac{21}{4}
\]

So,

\[
r = \sqrt{\frac{21}{4}} = \frac{\sqrt{21}}{2}
\]

### 2. \( a_2 = 9 \) and \( a_4 = 1 \)
Using the formula:

\[
a_2 = a_1 \cdot r = 9
\]
\[
a_4 = a_1 \cdot r^3 = 1
\]

Dividing the second by the first:

\[
\frac{a_4}{a_2} = \frac{1}{9} = r^2
\]

So,

\[
r = \sqrt{\frac{1}{9}} = \frac{1}{3}
\]

### 3. \( a_2 = 27 \) and \( a_4 = 81 \)
Using the formula:

\[
a_2 = a_1 \cdot r = 27
\]
\[
a_4 = a_1 \cdot r^3 = 81
\]

Divide:

\[
\frac{a_4}{a_2} = \frac{81}{27} = 3 = r^2
\]

So,

\[
r = \sqrt{3}
\]

### 4. \( a_3 = 81 \) and \( a_5 = 625 \)
Using the formula:

\[
a_3 = a_1 \cdot r^2 = 81
\]
\[
a_5 = a_1 \cdot r^4 = 625
\]

Dividing:

\[
\frac{a_5}{a_3} = \frac{625}{81} = \left(\frac{25}{9}\right) = r^2
\]

So,

\[
r = \frac{5}{3}
\]

### 5. \( a_2 = 64 \) and \( a_4 = 256 \)
Using the formula:

\[
a_2 = a_1 \cdot r = 64
\]
\[
a_4 = a_1 \cdot r^3 = 256
\]

Dividing:

\[
\frac{a_4}{a_2} = \frac{256}{64} = 4 = r^2
\]

So,

\[
r = 2
\]

### 6. \( a_1 = 5 \) and \( a_4 = 125 \)
Using the formula:

\[
a_1 = 5
\]
\[
a_4 = 5 \cdot r^3 = 125
\]

Solving for \( r \):

\[
r^3 = \frac{125}{5} = 25
\]

\[
r = \sqrt[3]{25}
\]

### 7. \( a_1 = 9 \) and \( a_4 = 16 \)
Using the formula:

\[
a_1 = 9
\]
\[
a_4 = 9 \cdot r^3 = 16
\]

Solving for \( r \):

\[
r^3 = \frac{16}{9}
\]

\[
r = \sqrt[3]{\frac{16}{9}}
\]

If you have any questions or need further explanations on any part, feel free to ask!

### Related Questions:
1. How do you find the first term of a geometric sequence given the common ratio and another term?
2. What are some applications of geometric sequences in real-world scenarios?
3. How do you find the sum of the first \( n \) terms of a geometric sequence?
4. What is the difference between geometric and arithmetic sequences?
5. How do you handle sequences when the common ratio is negative?

**Tip:** Always verify the common ratio by substituting it back into the terms to ensure the sequence makes sense!

Determine the common ratio of the geometric sequences given the pairs of terms below.

This problem focuses on finding the common ratio of geometric sequences using given terms. By applying the formula a_n = a_1 * r^(n-1), students can calculate the common ratio step by step.

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