To solve each question, let's use the properties of a geometric progression (GP).

---

Problem 71

Given:

• The 5th term of a GP is 50.
• The sum of the first 5 terms is 155.

Let:

• The first term be $a$.
• The common ratio be $r$.

For a GP:

1. The $n$-th term is given by: $T_n = a \cdot r^{n-1}$

   For the 5th term: $T_5 = a \cdot r^4 = 50$

2. The sum of the first $n$ terms is: $S_n = a \frac{r^n - 1}{r - 1}$

   For the sum of the first 5 terms: $S_5 = a \frac{r^5 - 1}{r - 1} = 155$

Using these equations:

1. From the 5th term: $a \cdot r^4 = 50$

2. From the sum of the first 5 terms: $a \frac{r^5 - 1}{r - 1} = 155$

By solving these two equations simultaneously, we find that: $r = 2$

Thus, the answer is: (b) 2

---

Problem 72

Given:

• The sum of the first 8 terms of a GP is 510.
• The first term is 2.

Let:

• The common ratio be $r$.

For a GP:

1. The sum of the first $n$ terms: $S_n = a \frac{r^n - 1}{r - 1}$

For the sum of the first 8 terms: $S_8 = 2 \frac{r^8 - 1}{r - 1} = 510$

Solving this equation yields: $r = 3$

Thus, the answer is: (b) 3

---

Would you like a more detailed solution for each step?

---

Related Questions:

1. How do you determine the sum of a finite geometric series if you know the first term and the common ratio?
2. How does changing the common ratio affect the behavior of a geometric series?
3. How do you find the nth term in a geometric sequence when only given the first term and the sum?
4. How can you solve for the common ratio if the sum of terms is given but the sequence length is unknown?
5. What happens to the sum of terms in a geometric series if the common ratio is between 0 and 1?

Tip: In geometric progressions, the sum of terms can grow quickly if the common ratio is greater than 1, making it useful for modeling exponential growth situations.