Let a₁,a₂,a₃,... be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then a₁a₉+a₂a₄a₉+a₅+a₇ is equal to __________.

Solution

Maths sir · 4 steps

Pin down the middle term a₅ first, then rewrite every product as a power of a₅.

1. Find the fifth term. In a GP, a₄ a₆ = a₅² because a₄ and a₆ sit equally far from a₅. So a₅² = 9, and since all terms are positive, a₅ = 3.
2. Find the seventh term. From a₅ + a₇ = 24 we get a₇ = 21. This gives r² = 21/3 = 7, so r = √7 > 1, which fits an increasing GP.
3. Rewrite the products. With aₙ = a₁ rⁿ⁻¹, the first product is a₁ a₉ = a₁² r⁸ = (a₁ r⁴)² = a₅² = 9. The second is a₂ a₄ a₉ = a₁³ r¹⁺³⁺⁸ = a₁³ r¹² = (a₁ r⁴)³ = a₅³ = 27.

   a₁a₉ + a₂a₄a₉ + a₅ + a₇ = 9 + 27 + 3 + 21

4. So. The required value is 60.

The idea to keep. In a GP, a product of terms whose indices average to k equals aₖ raised to the number of terms. Check the index sum: 2 + 4 + 9 = 15 = 3 × 5, so a₂a₄a₉ = a₅³.

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