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!