Let's analyze and solve the given problems from the image:

1. For each of the following GPs, find the common ratio $r$.

A geometric progression (GP) has a constant ratio between consecutive terms. The common ratio $r$ can be calculated by dividing the second term by the first term.

(a) $3, 9, 27, \dots$
$r = \frac{9}{3} = 3$

(b) $10, 5, \frac{5}{2}, \dots$
$r = \frac{5}{10} = \frac{1}{2}$

(c) $3\sqrt{3}, 3\sqrt{6}, 6\sqrt{3}, \dots$
$r = \frac{3\sqrt{6}}{3\sqrt{3}} = \frac{\sqrt{6}}{\sqrt{3}} = \sqrt{2}$

(d) $x + x^2, x + x^3, x + x^4, \dots$
$r = \frac{x + x^3}{x + x^2} = \frac{x(1 + x^2)}{x(1 + x)} = \frac{1 + x^2}{1 + x}$

2. Find the indicated terms of these GPs.

The general term for a geometric progression is $T_n = ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

(a) $2, 10, 50, \dots$

• First term $a = 2$
• Common ratio $r = \frac{10}{2} = 5$

Now, let's find the 7th term:
$T_7 = ar^{7-1} = 2 \times 5^6 = 2 \times 15625 = 31250$

(b) $\frac{1}{2}, 2, 8, \dots$

• First term $a = \frac{1}{2}$
• Common ratio $r = \frac{2}{\frac{1}{2}} = 4$

Find the 8th term:
$T_8 = ar^{8-1} = \frac{1}{2} \times 4^7 = \frac{1}{2} \times 16384 = 8192$

3. Find the fifth term of the GP $\frac{1}{27}, \frac{9}{81}, 3, \dots$. How many terms are there?

First, find the common ratio: $r = \frac{\frac{9}{81}}{\frac{1}{27}} = \frac{9}{81} \times \frac{27}{1} = \frac{9 \times 27}{81} = 3$

Now, find the 5th term:
$T_5 = ar^{5-1} = \frac{1}{27} \times 3^4 = \frac{1}{27} \times 81 = 3$

To find the total number of terms, note that the last term is 81, which is written as: $T_n = ar^{n-1} = \frac{1}{27} \times 3^{n-1} = 81$ Solve for $n$:
$\frac{1}{27} \times 3^{n-1} = 81$ $3^{n-1} = 81 \times 27 = 2187 = 3^7$ $n - 1 = 7 \quad \Rightarrow \quad n = 8$

4. In a GP, the fourth term is 18 and the seventh term is 486. Find the second term.

Let the first term be $a$ and the common ratio be $r$.

• Fourth term: $T_4 = ar^3 = 18$
• Seventh term: $T_7 = ar^6 = 486$

Dividing the second equation by the first to eliminate $a$:
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