Suppose `xa n dy` are two real numbers such that the rth mean between `xa n d2y` is equal to the rth mean between `2xa n dy` when `n` arithmetic means are inserted between them in both the cases. Show that `(n+1)/r-y/x=1.`

Explanation

Series I-`X,A_1,A_2...A_r...A_n,2y`

`A_r=X+(r+1-1)alpha`

`A_r=X+r((2y-X)/(n+1))-(1)`

`2y=X+(n+2-1)d`

`d=(2y-X)/(n+1)`

Series 2-`2X,A_1,A_2,...,A_r,....A_n,y`

`A_r=2X+(r+1-1)d`

`A_r=2x+r((y-2X)/(n+1))-(2)`

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