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Question

If a,b,c and d are in geometric sequence, then prove that (bc)2+(ca)2+(db)2=(ad)2

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Solution

Let first term be a and common ratio r.
b=ar,c=ar2,d=ar3
L.H.S. =(bc)2+(ca)2+(db)2
=(arar2)2+(ar2a)2+(ar3ar)2
=a2(rr2)2+a2(r21)2+a2(r3r)2
=a2[(rr2)2+(r21)2+(r3r)2]
=a2(r22r3+r4+r42r3+1+r62r4+r2)
=a2(r62r3+1)
=a2(1r3)2 .... (1)
(ad)2=(aar3)2
=a2(1r3)2 ....(2)
From (1) and (2), we have
(bc)2+(ca)2+(db)2=(ad)2

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