Question -
Answer -
Given,┬аa,┬аb,┬аc,┬аd┬аarein G.P.
So, we have
bc┬а=┬аad┬а тАж (1)
b2┬а=┬аac┬а┬атАж (2)
c2┬а=┬аbd┬а тАж (3)
Taking the R.H.S. wehave
R.H.S.
= (ab┬а+┬аbc┬а+┬аcd)2
= (ab┬а+┬аad┬а+┬аcd)2┬а[Using (1)]
= [ab┬а+┬аd┬а(a┬а+┬аc)]2
=┬аa2b2┬а+2abd┬а(a┬а+┬аc) +┬аd2┬а(a┬а+┬аc)2
=┬аa2b2┬а+2a2bd┬а+2acbd┬а+┬аd2(a2┬а+ 2ac┬а+┬аc2)
=┬аa2b2┬а+2a2c2┬а+ 2b2c2┬а+┬аd2a2┬а+2d2b2┬а+┬аd2c2┬а[Using (1) and (2)]
=┬аa2b2┬а+┬аa2c2┬а+┬аa2c2┬а+┬аb2c2┬а+┬аb2c2┬а+┬аd2a2┬а+┬аd2b2┬а+┬аd2b2┬а+┬аd2c2
=┬аa2b2┬а+┬аa2c2┬а+┬аa2d2┬а+┬аb2┬а├Ч┬аb2┬а+┬аb2c2┬а+┬аb2d2┬а+┬аc2b2┬а+┬аc2┬а├Ч┬аc2┬а+┬аc2d2
[Using(2) and (3) and rearranging terms]
=┬аa2(b2┬а+┬аc2┬а+┬аd2)+┬аb2┬а(b2┬а+┬аc2┬а+┬аd2)+┬аc2┬а(b2+┬аc2┬а+┬аd2)
= (a2┬а+┬аb2┬а+┬аc2)(b2┬а+┬аc2┬а+┬аd2)
= L.H.S.
Thus, L.H.S. = R.H.S.
Therefore,
(a2┬а+b2┬а+ c2)(b2┬а+ c2┬а+ d2)= (ab + bc + cd)2