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Question
If x, y and z are in continued proportion, Prove that:
`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`
Solution
`x/y = y/z`
`\implies` y2 = xz
LHS = `x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2)`
= `(x^3 + y^3 + z^3)/(x^2.y^2z^2)`
`(x^3 + y^3 + z^3)/(x^3z^3) = x^3/(x^3z^3) + y^3/(x^3z^3) + z^3/(x^3z^3)`
= `1/z^3 + y^3/y^6 + 1/x^3`
= `1/z^3 + 1/y^3 + 1/x^3`
= RHS
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