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KVPY2019MathematicsSequences and Series

The number of three digit numbers a b c ¯ such that the arithmetic mean of b and c and the square of their geometric mean are equal is

Options

  1. A9
  2. B18
  3. C36
  4. D54

Correct answer

B. 18

Step-by-step solution

It is given that, the number of three digit nụmber a b c ¯ , such that b + c 2 = b c the above relation is true if b = c = 0 And if neither b nor c is zero, then 1 b + 1 c = 2 and b , c ∈ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 Then b = c = 1 and a ∈ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 So, total number of such three digit number are 2 × 9 = 18

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