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Question

The harmonic mean of 2 numbers is 4. Their arithmetic mean A and geometric mean G satisfy the relation 2A + G2 = 27. The sum of the numbers is __


Solution

Let the numbers be a & b

H = 4

G2  = AH = 4A         -------------(1)

Given 2A +  G2 = 27    -----------(2)

From (1) & (2)

6A = 27 A =  92

  G2  = 4 ×  92 = 18

If we know A and G, the 2 numbers are given by,

a, b = A ±  A2G2

        = 92 ± 81418

        = 92  ± 32 = 6, 3 (or) 3, 6

Therefore, the sum of the numbers = 9

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