In an increasing geometric series, the sum of the second and the sixth term is 25/2 and the product of the third and fifth term is 25. Then, the sum of 4ᵗʰ, 6ᵗʰ and 8ᵗʰ terms is equal to :

Solution

Maths sir · 4 steps

The 4th term sits in the middle of everything here, so find it first and write every other term around it.

1. Find the middle term. Let the first term be a and the ratio r. Then T₃ T₅ = ar² · ar⁴ = (ar³)² = T₄² = 25, so T₄ = ± 5. If T₄ = -5, then T₂ + T₆ = -5(1/r² + r²) is negative and cannot equal 25/2. So T₄ = 5.
2. Use the sum. Write the other terms around T₄: T₂ = 5/r² and T₆ = 5r². Their sum is 25/2, so divide both sides by 5.

   r² + 1/r² = 5/2 ⇒ 2r⁴ - 5r² + 2 = 0 ⇒ r² = 2 or 1/2

3. Pick the ratio. The terms are positive and the series is increasing, so r > 1. That gives r² = 2.
4. So. T₄ + T₆ + T₈ = 5 + 5r² + 5r⁴ = 5(1 + 2 + 4) = 35, option (d).

The idea to keep. When the given terms are placed evenly around one term, write them as T/rᵏ and Trᵏ. A product of such a pair gives T², and a sum of such a pair gives an equation in r² alone.

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