Given: SUPPOSE X and y are two real numbers such that the rth mean between X and 2Y is equal to the rth mean between 2x and y.

To find: When n AMs are inserted between them in both CASES prove that  N+1/r - Y/x = 1.

Solution:

• Now we have given the two series:

• Series 1: x , ........., 2y

• Adding n terms:

                   x ,A1, A2, A3, ............An, 2y

• Now 2y = x + (n+2-1)d

                   d = 2y - x / n + 1

                   Ar = x + (r+1-1)d

                   Ar = x + r(2y-x / n+1) ...............(i)

• Series 2: 2x , .......... y

• Adding n terms:

                   2x, A1, A2, A3, ............An, y

• Now y = 2x + (n+2-1)d

                   d = y - 2x / n + 1

                   Ar = 2x + (r+1-1)d

                   Ar = 2x + r(y-2x / n+1) ...............(ii)

• Now (i) and (ii) are equal so:

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

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

                   x = r/n+1 (x + y)

                   n+1 / r = y+x / x

                   n+1 / r = y/x + 1

• Hence proved.

Answer:

          So we proved that n+1 / r = y/x + 1.