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 :

Solution

Maths sir · 4 steps

Both sums share the factor 1 + r + r², so divide one by the other and r comes out at once.

1. Write both sums. With first term a and ratio r, the second, third and fourth terms add to ar(1 + r + r²) = 3. The sixth, seventh and eighth terms add to ar⁵(1 + r + r²) = 243.
2. Divide to get r. Dividing the second sum by the first cancels a and 1 + r + r², leaving r⁴ = 81. The terms are positive, so r = 3.
3. Find a. Put r = 3 in the first sum: a × 3 × (1 + 3 + 9) = 39a = 3, so a = 1/13.

   S₅₀ = a(r⁵⁰ - 1)/(r - 1) = 1/13 × (3⁵⁰ - 1)/2 = 1/26(3⁵⁰ - 1)

4. So. The sum of the first 50 terms is 1/26(3⁵⁰ - 1), which is option (d).

The idea to keep. When two groups of the same number of consecutive G.P. terms are given, divide them. The common bracket cancels and you are left with a pure power of r.

Answer from the NTA final answer key. Solution by PrepWiser.