In an increasing geometric progression of positive terms, the sum of the second and sixth terms is 70/3 and the product of the third and fifth terms is 49. Then the sum of the 4ᵗʰ, 6ᵗʰ and 8ᵗʰ terms is equal to:

Solution

Maths sir · 4 steps

The product T₃ T₅ hands you the middle term T₄ straight away, so build everything around it.

1. Find the fourth term. Write the terms as Tₙ = arⁿ⁻¹. Then T₃ T₅ = ar² · ar⁴ = (ar³)² = T₄². So T₄² = 49, and since every term is positive, T₄ = 7.
2. Use the sum. Measure the other terms from T₄: T₂ = 7/r² and T₆ = 7r². Their sum is 70/3, so dividing by 7 gives the equation below.

   r² + 1/r² = 10/3 ⇒ 3r⁴ - 10r² + 3 = 0 ⇒ r² = 3 or 1/3

3. Pick the ratio. The progression is increasing with positive terms, so r > 1. That rules out r² = 1/3 and leaves r² = 3.
4. So. T₄ + T₆ + T₈ = 7 + 7r² + 7r⁴ = 7(1 + 3 + 9) = 7 × 13 = 91, which is option (c).

The idea to keep. In a GP, terms placed symmetrically around a middle term multiply to the square of that term. Write the other terms as that middle term times or divided by powers of r.

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