Question
Wed May 30, 2012 By: Vikrant
 

Find the relation between x and y such that the rth mean between x and 2y may be same as the rth mean between 2x and y, if n means are inserted in each case.

Expert Reply
Wed May 30, 2012

Let x, A1, A2, … An, 2y be in AP.

Then common difference = (2y - x)/ (n + 1)

Ar = x + rd = x + r (2y - x)/ (n + 1) = [x (n + 1) + r(2y - x)]/ (n + 1)

Now 2x, B1, B2, … Bn, y are in AP.

Then common difference = (y - 2x)/ (n + 1)

Br = x + rd = x + r (y - 2x)/ (n + 1) = [x (n + 1) + r(y - 2x)]/ (n + 1)

Since, Ar = Br, we have:

[x (n + 1) + r(2y - x)]/ (n + 1) = [x (n + 1) + r(y - 2x)]/ (n + 1)

On simplifying, we get,

ry = x(n - r + 1)

This is the required relation.

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