Le x be the arithmetic mean and y, z be the two geometric means between any two positive numbers, then y^3+z^3 x y z = -----------

Step-by-step solution

Let the two positive numbers be a and b . The arithmetic mean x is given by x = a+b 2 . The two geometric means y and z between a and b imply that a, y, z, b are in geometric progression. Let the common ratio be r . Then y = ar , z = ar^2 , and b = ar^3 . From b = ar^3 , we have r = ( b a )^ 1/3 . The product y z = (ar)(ar^2) = a^2 r^3 = a^2 ( b a ) = ab . The sum y^3 + z^3 = (ar)^3 + (ar^2)^3 = a^3 r^3 + a^3 r^6 = a^3 ( b a ) + a^3 ( b a )^2 = a^2 b + a b^2 = ab(a+b) . Substituting these into the expression y^3+z^