I can help extract the math problems from your image and solve them. Let me first transcribe the relevant parts of the image.

1. Problem 6:
   In a GP where terms are all positive, the third term exceeds the first term by 32, while the fifth term exceeds the first term by 320. Calculate the first term.

2. Problem 7:
   Find the geometric means of the following:
   (a) $3, 12$
   (b) $4, 36$
   (c) $5, 20, 45$

3. Problem 8:
   Find the indicated sums of these GPs:
   (i) $5, 10, 20, \dots$ find $S_6$ and $S_8$.
   (ii) $3, -6, 12, \dots$ find $S_5$ and $S_6$.

4. Problem 9:
   Given $a + a r + a r^2 + a r^3 = 40$, then the first positive term of a GP, calculate the sum of the first 6 terms of the GP.

5. Problem 11:
   The sum of the first 3 terms of a GP is 117, and the sum of the next 3 terms is 36. Calculate the third term.

I will begin solving these one by one. Let's start with Problem 6.

Problem 6:

Given that:

• The third term exceeds the first term by 32, i.e., $T_3 = a + 32$.
• The fifth term exceeds the first term by 320, i.e., $T_5 = a + 320$.

The general formula for the $n$-th term of a geometric progression (GP) is: $T_n = a \cdot r^{n-1}$

For the third term: $T_3 = a \cdot r^2 = a + 32$

For the fifth term: $T_5 = a \cdot r^4 = a + 320$

Now, I will solve this system of equations to find $a$ (the first term) and $r$ (the common ratio). Let's compute.The solutions for the first term $a$ and the common ratio $r$ are:

• $a = 4$, $r = -3$
• $a = 4$, $r = 3$

Since the terms of the GP are stated to be positive, we discard the solution with $r = -3$. Therefore, the first term is $a = 4$ and the common ratio is $r = 3$.

Would you like to proceed with solving the next problem, or do you have any questions on this solution?

Here are some relative questions to expand on this problem:

1. What is the formula for the $n$-th term of a geometric progression?
2. How does the common ratio affect the growth of a geometric progression?
3. What is the sum of the first 5 terms of this GP?
4. What happens to the sequence if the common ratio is negative?
5. How do we find the sum of infinite terms in a geometric progression?

Tip: In a GP, if the common ratio is greater than 1, the terms grow rapidly.