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Let x be the arithmetic mean and y ,z be...

Let `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)=dot` (1997C, 2M)

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The correct Answer is:
2
Let the two positive numbers be a and b
`:. X = (a +b)/(2)` [since, x is AM between a and b]..(i)
and `(a)/(y) = (y)/(z) = (z)/(b)` [since, y , z are GM's between a and b]
`:. a = (y^(2))/(z) and b = (z^(2))/(y)`

On substituting the values of a and b in Eq. (i), we get
`2x = (y^(2))/(z) + (z^(2))/(y)`
`rArr (y^(3) + z^(3))/(yz) = 2x`
`rArr (y^(3) + z^(3))/(xyz) = 2`
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