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 eighth terms is 243, then the sum of the first 50 terms of this G.P. is:

Let the first term of the required geometric progression be $$a$$ and let its common ratio be $$r$$. Then the successive terms are $$a,\; ar,\; ar^{2},\; ar^{3},\; \dots$$

We are told that the sum of the second, third and fourth terms equals $$3$$. Writing that out, we have

$$ar + ar^{2} + ar^{3} = 3.$$

Factoring out the common term $$ar$$ gives

$$ar(1 + r + r^{2}) = 3. \quad -(1)$$

Next, the sixth, seventh and eighth terms of the same G.P. are $$ar^{5},\; ar^{6},\; ar^{7}$$, and their sum is given to be $$243$$. Hence

$$ar^{5} + ar^{6} + ar^{7} = 243.$$

Again factoring, we obtain

$$ar^{5}(1 + r + r^{2}) = 243. \quad -(2)$$

Now we divide equation (2) by equation (1). On the left-hand side the common factor $$(1 + r + r^{2})$$ as well as $$a$$ cancel out, leaving

$$\frac{ar^{5}(1 + r + r^{2})}{ar(1 + r + r^{2})} = r^{4}.$$

On the right-hand side we have

$$\frac{243}{3} = 81.$$

Equating the two results gives

$$r^{4} = 81.$$

Since all terms are positive, we take the positive fourth root, yielding

$$r = 3.$$

We now substitute $$r = 3$$ back into equation (1) to determine $$a$$. That equation becomes

$$a \cdot 3 \, (1 + 3 + 9) = 3.$$

The bracket simplifies to $$1 + 3 + 9 = 13$$, so we have

$$3a \times 13 = 3$$

or

$$39a = 3,$$

which gives

$$a = \frac{3}{39} = \frac{1}{13}.$$

With $$a = \dfrac{1}{13}$$ and $$r = 3$$ in hand, we can now find the sum of the first $$50$$ terms. The standard formula for the sum of the first $$n$$ terms of a G.P. with first term $$a$$ and common ratio $$r \neq 1$$ is

$$S_{n} = a \,\frac{r^{n} - 1}{r - 1}.$$

Applying this formula with $$n = 50$$, $$a = \dfrac{1}{13}$$ and $$r = 3$$, we obtain

$$S_{50} = \frac{1}{13}\,\frac{3^{50} - 1}{3 - 1}.$$

The denominator $$3 - 1$$ equals $$2$$, so we simplify to

$$S_{50} = \frac{1}{13}\,\frac{3^{50} - 1}{2} = \frac{1}{26}\,(3^{50} - 1).$$

Hence, the correct answer is Option B.