Let a₁, a₂, a₃, .... be a G.P. of increasing positive numbers. Let the sum of its 6ᵗʰ and 8ᵗʰ terms be 2 and the product of its 3ʳᵈ and 5ᵗʰ terms be 1/9. Then 6(a₂+a₄)(a₄+a₆) is equal to

Solution

Maths sir · 3 steps

Write every term in terms of a₄: the product gives a₄ and the sum gives r².

1. Find the middle term. In a G.P., a₃ a₅ = a₄², so a₄² = 1/9. The terms are positive, so a₄ = 1/3.
2. Find the ratio. Write a₆ + a₈ = a₄(r² + r⁴) = 2. This gives r⁴ + r² - 6 = 0, that is (r² + 3)(r² - 2) = 0. Since r² cannot be negative, r² = 2. The G.P. is increasing, so r = √2 > 1.

   a₂ + a₄ = a₄(1/r² + 1) = 1/3 × 3/2 = 1/2, a₄ + a₆ = a₄(1 + r²) = 1/3 × 3 = 1

3. So. 6(a₂ + a₄)(a₄ + a₆) = 6 × 1/2 × 1 = 3, which is option (d).

The idea to keep. In a G.P., the product of two terms equally spaced around a middle term equals the square of that middle term. Express every other term through that middle term and r².

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