Final answer:
The sum of the first 50 terms of the G.P. is found by determining the first term and the common ratio, using the given sums of the second to fourth and sixth to eighth terms. After calculating these values, we can use the formula for the sum of the first 'n' terms of a G.P. to get the desired sum. Option d. 1/26 (3⁵⁰-1) is the correct answer.
Step-by-step explanation:
The question involves determining the sum of the first 50 terms of a geometric progression (G.P.) given that the sum of the second, third, and fourth terms is 3, and the sum of the sixth, seventh, and eighth terms is 243. We can denote the first term of the G.P. as a and the common ratio as r. The sum of the first n terms of a G.P. is given by Sn = a(1 - rn)/(1 - r) for r ≠ 1.
For the sum of the second, third, and fourth terms: ar + ar2 + ar3 = 3, which simplifies to ar(1 + r + r2) = 3. For the sum of the sixth, seventh, and eighth terms: ar5 + ar6 + ar7 = 243, simplifying to ar5(1 + r + r2) = 243. We can divide the second equation by the first to find the common ratio r.
Once r is known, we find the first term a from one of the equations and then compute the sum of the first 50 terms S50. The correct answer turns out to be 1/26 (350 - 1), which corresponds to option d.