Wed August 15, 2012 By:

if x,y,z are in ap ,x,xy,z are in a gp show that x,x^2(square)y,z are in hp

Expert Reply
Mon August 20, 2012

Given, x,y,z are in A.P. 
Therefore, x+z = 2y ......(1)
Also, x,xy,z are in G.P
Therefore, (xy)2 = xz
x2y2 = xz
xy2 = z ....(2)

Next, we have to prove that x,x2y,z are in H.P
Now, 1/x+1/z = (z+x)/xz
= 2y/x.xy2              [Using (1) and (2)]
= 2/x2y
Therefore, 1/x+1/z =2/x2y
Therefore,x,x2y,z are in H.P 

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