If the sum of the second , third and fourth terms of a positive term G.P is 3 and the sum of its sixth , seventh and eight terms is 243, then the sum of the first 50 terms of this G.P is :

Options

A. `1/26 (3^49 - 1)`
B. `1/26 (3^50 - 1)`
C. `2/13 (3^50 - 1)`
D. `1/13 (3^50 - 1)`

Answer

To solve the problem, we need to find the sum of the first 50 terms of a geometric progression (G.P.) given certain conditions about the terms of the G.P.

### Step-by-Step Solution:

1. **Define the G.P.**:
Let the first term of the G.P. be \( a \) and the common ratio be \( r \). The terms of the G.P. can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- Fifth term: \( ar^4 \)
- Sixth term: \( ar^5 \)
- Seventh term: \( ar^6 \)
- Eighth term: \( ar^7 \)

2. **Set up the equations based on the given conditions**:
- The sum of the second, third, and fourth terms is given as:
\[ ar + ar^2 + ar^3 = 3 \]
Factoring out \( ar \):
\[ ar(1 + r + r^2) = 3 \quad \text{(1)} \]

- The sum of the sixth, seventh, and eighth terms is given as:
\[ ar^5 + ar^6 + ar^7 = 243 \]
Factoring out \( ar^5 \):
\[ ar^5(1 + r + r^2) = 243 \quad \text{(2)} \]

3. **Divide equation (2) by equation (1)**:
\[ \frac{ar^5(1 + r + r^2)}{ar(1 + r + r^2)} = \frac{243}{3} \]
This simplifies to:
\[ r^4 = 81 \]
Taking the fourth root:
\[ r = 3 \quad \text{(since \( r \) is positive)} \]

4. **Substitute \( r \) back into equation (1)**:
Substitute \( r = 3 \) into equation (1):
\[ ar(1 + 3 + 3^2) = 3 \]
Simplifying:
\[ ar(1 + 3 + 9) = 3 \implies ar \cdot 13 = 3 \implies ar = \frac{3}{13} \quad \text{(3)} \]

5. **Find \( a \)**:
Since \( r = 3 \):
\[ a \cdot 3 = \frac{3}{13} \implies a = \frac{1}{13} \]

6. **Calculate the sum of the first 50 terms of the G.P.**:
The sum of the first \( n \) terms of a G.P. is given by:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
For \( n = 50 \):
\[ S_{50} = \frac{\frac{1}{13}(3^{50} - 1)}{3 - 1} = \frac{\frac{1}{13}(3^{50} - 1)}{2} \]
Simplifying:
\[ S_{50} = \frac{3^{50} - 1}{26} \]

### Final Answer:
The sum of the first 50 terms of the G.P. is:
\[ S_{50} = \frac{3^{50} - 1}{26} \]