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The value of n for which (x^(n+1)+y^(n...

The value of n for which ` (x^(n+1)+y^(n+1))/(x^(n)+y^(n))` is the geometric mean of x nd y is

A

`n =-1/2`

B

`n=1/2`

C

`n =1`

D

`n =-1`

Text Solution

Verified by Experts

The correct Answer is:
a
` (x^(n+1)+y^(n+1))/(x^(n)+y^(n))=sqrt(xy)`
` rArr x^(n+1) +y^(n+1) = (xy)^(1/2)(x^(n)+y^(n))`
` rArr x^(n) * x+y^(n) y = x^(n) * x^(1/2)y^(1/2)+y^(n)*x^(1/2) y^(1/2)`
`rArr x^(n) (x - sqrt(xy)) +y^(n)(y-sqrt(xy))=0`
`rArr x^(n) * sqrt(x)(sqrt(x)-sqrt(y))+sqrt(y)*y^(n)(sqrt(y)-sqrt(x))=0`
` rArr (sqrt(x)-sqrt(y)) (x^(n)*sqrt(x)-y^(n)*sqrt(y))=0`
For `x!= y`,
` rArr (x/y)^(n+1/2)=1rArr n+1/2 = 0 rArr n = -1/2`
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