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2015-04-15T16:05:06+05:30

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Let the two real number be  a and  b.  Let  b >= a.

AM =  (a+b)/2                        GM = √(a b)

AM² = (a² + b² + 2 a b) / 4            GM² = a b

now find :
  4  AM²  -  4 GM² =    a² + b² + 2 a b  - 4 a b
                       =    a² + b² - 2 a b
                        =  ( a - b )²          ≥  0
      as it is a square of a real number, it is always  greater than or equal to 0.

   Hence  AM²  ≥  GM²
                  | AM |   ≥    | GM  |

Thus the magnitude of arithmetic mean is greater than or equal to the magnitude of  geometric mean.

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