Let a₁, a₂, a₃, …. be a G.P. of increasing positive numbers. If a₃ a₅=729 and a₂+a₄=111/4, then 24(a₁+a₂+a₃) is equal to

Solution

Maths sir · 4 steps

Spot that a₃ and a₅ sit on either side of a₄, so their product is a₄², and the rest falls out.

1. Use the middle term. In a G.P., two terms equally far from a middle term multiply to the square of that middle term. So a₃ a₅ = a₄² = 729, which gives a₄ = 27. It is positive because all the terms are positive.
2. Find the ratio. From a₂ + a₄ = 111/4 we get a₂ = 111/4 - 27 = 3/4. Then r² = a₄/a₂ = 27/(3/4) = 36, so r = 6, since the G.P. is increasing and positive.
3. First three terms. a₁ = a₂/r = 1/8, a₂ = 3/4 = 6/8 and a₃ = a₂ r = 9/2 = 36/8. Their sum is (1 + 6 + 36)/8 = 43/8.

   24(a₁+a₂+a₃) = 24 × 43/8 = 3 × 43 = 129

4. So. The value is 129, which is option (b).

The idea to keep. In a G.P., aₙ₋ₖ aₙ₊ₖ = aₙ². When a product of two terms is given, first look for the term midway between them. It usually gives you one term directly.

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