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n arlithmetic means are inserted between...

`n` arlithmetic means are inserted between `xa n d2y` and then between `2xa n dydot` If the rth means in each case be equal, then find the ratio `x//ydot`

Text Solution

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The correct Answer is:
`(r)/(n-r+1)`
`a,x_(1),x_(2),…,x_(n),b` are in A.P. Hence,
`b=T_(n+2)=a+(n+1)d`
or `(b-a)/(n+1)=d`
`rArrx_(r)=T_(r+1)=a+rd`
`=a+(r(b-a))/(n+1)`
`=(a(n-r+1)+rb)/(n+1)`
Put a=x and b=2y and then again put a=2x and b=y and equate the results as the two means are equal. Then
`(x(n-r+1)+2yr)/(n+1)=(2x(n-r+1)+yr)/(n+1)`
`rArrx(n-r+1)=yr`
`rArrx/y=r/(n-r+1)`
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