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Mathematics

If y is the mean proportional between x and z, prove that

xyz(x + y + z)3 = (xy + yz + zx)3

Ratio Proportion

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Answer

Given, y is the mean proportional between x and z then,

y2 = xz    [….Eq 1]

Given, xyz(x + y + z)3 = (xy + yz + zx)3

Solving L.H.S. first,

   xyz(x + y + z)3
= xz.y(x + y + z)3
Putting value of xz as y2 from equation 1:
= y2.y(x + y + z)3
= y3(x + y + z)3
= (y(x + y + z))3
= (xy + y2 + yz)3
= (xy + xz + yz)3 = R.H.S.

L.H.S. = R.H.S. , hence proved, xyz(x + y + z)3 = (xy + yz + zx)3.

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