If a₁, a₂, a₃ ...... and b₁, b₂, b₃ ....... are A.P., and a₁ = 2, a₁₀ = 3, a₁b₁ = 1 = a₁₀b₁₀, then a₄ b₄ is equal to -

Solution

Maths sir · 4 steps

Treat the two A.P.s separately: find a₄ and b₄ on their own, then multiply.

1. First A.P. From a₁ = 2 to a₁₀ = 3 there are 9 equal steps, so the common difference is (3-2)/9 = 1/9. Then a₄ = 2 + 3 × 1/9 = 7/3.
2. Ends of second A.P. From a₁ b₁ = 1 we get b₁ = 1/2. From a₁₀ b₁₀ = 1 we get b₁₀ = 1/3.
3. Second A.P. Its common difference is (1/3-1/2)/9 = (-1/6)/9 = -1/54. Then b₄ = 1/2 - 3 × 1/54 = 1/2 - 1/18 = 4/9.

   a₄ b₄ = 7/3 × 4/9 = 28/27

4. So. a₄ b₄ = 28/27, which is option (d).

The idea to keep. In an A.P., the nth term is a₁ + (n-1)d, and d comes from any two known terms. A product of two A.P.s is not an A.P., so find each factor first.

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