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KVPY2017MathematicsSequences and Series

Let A, G and H be the arithmetic mean, geometric mean and harmonic mean, respetively of two distinct positive real numbers. If is the smallest of the two roots of the equation A ( G - H ) x ²+ G ( H - A ) x + H ( A - G )=0 , then

Options

  1. A-2 < < -1
  2. B0 < < 1
  3. C-1 < < 0
  4. D1 < < 2

Correct answer

B. 0 < < 1

Step-by-step solution

x =1 is a root as sum of coefficient =0 Now = H(A-G) A(G+H) Put =1 aligned &= H A-H G A G-A H = G²-H G A G-A H &= G(G-H) A(G-H) = G A G.M. ] aligned

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