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If p , g₁ , g₂ and q are in GP and m is the arithmetic mean of p and q , then g₁^2 g₂ + g₂^2 g₁ is equal to

Options

  1. Am
  2. B2m
  3. C1
  4. D1 2

Correct answer

B. 2m

Step-by-step solution

Given p , g₁ , g₂ , q are in GP. Let the common ratio of the GP be r . Then g₁ = pr , g₂ = pr^2 , and q = pr^3 . The arithmetic mean of p and q is m , so m = p + q 2 p + q = 2m . Consider the expression g₁^2 g₂ + g₂^2 g₁ . Substituting the values of g₁ and g₂ : g₁^2 g₂ + g₂^2 g₁ = (pr)^2 pr^2 + (pr^2)^2 pr = p^2r^2 pr^2 + p^2r^4 pr = p + pr^3 Since q = pr^3 , we get: = p + q Substituting p + q = 2m : g₁^2 g₂ + g₂^2 g₁ = 2m Answer: 2m

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