Question

If x, y, z are in continued proportion, prove that: x²y²z²(x⁻⁴ + y⁻⁴ + z⁻⁴) = y⁻²(x⁴ + y⁴ + z⁴)

Answer 1

x, y, z are in continued proportion.

⇒ `x/y = y/z`

⇒ y² = xz

L.H.S.

= x²y²z²(x⁻⁴ + y⁻⁴ + z⁻⁴)

= x²y²z²x⁻⁴ + x²y²z²y⁻⁴ + x²y²z²z⁻⁴

= x⁻²y²z² + x²y⁻²z² + x²y²z⁻²

Substitute the condition y² = xz into each term:

= x⁻²(xz)z² + x²`(1/(xz))`z² + x²(xz)z⁻²

= `x^-1z^3 + xz + x^3/z`

= `(z^4 + x^2z^2 + x^4)/(xz)`

= `(x^4 + y^4 + z^4)/y^2`

= y⁻²(x⁴ + y⁴ + z⁴)

∴ L.H.S. = R.H.S.

Hence, proved.