Let x₁, x₂, x₃, x₄ be in a geometric progression. If 2,7,9,5 are subtracted respectively from x₁, x₂, x₃, x₄, then the resulting numbers are in an arithmetic progression. Then the value of 1/24(x₁ x₂ x₃ x₄) is:

Solution

Maths sir · 4 steps

Write the GP as a, ar, ar², ar³, then use the AP middle-term rule twice and divide.

1. Set up. Take x₁, x₂, x₃, x₄ = a, ar, ar², ar³. After subtracting, the numbers a-2, ar-7, ar²-9, ar³-5 are in AP.
2. Two AP conditions. In an AP each middle term is the average of its neighbours. From the first three terms, 2(ar-7) = (a-2)+(ar²-9). From the last three terms, 2(ar²-9) = (ar-7)+(ar³-5). Rearranging each one gives a perfect square (r-1)².

   a(r-1)² = -3, ar(r-1)² = -6

3. Divide. Dividing the second equation by the first gives r = 2. Then a(1)² = -3, so a = -3. The GP is -3, -6, -12, -24, and subtracting gives -5, -13, -21, -29, an AP with common difference -8.
4. So. x₁x₂x₃x₄ = (-3)(-6)(-12)(-24) = 5184, and 5184/24 = 216, which is option (b).

The idea to keep. For a GP written as a, ar, ar², …, two equations with a common factor are solved by dividing them, which gives r at once. Always check the final terms against the AP condition.

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