Question

find the relation between x and y in order that the rth mean between x and 2y maybe the same as the rth mean between 2x and y,n means being inserted in each case​

Answer 1

let x, a1, a2, an, 2y be in ap.

then commerce 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.