if a,b,c,d are in geometric sequence then prove that b-c^2+c-a^2+d-b^2=d-a^2
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Clearly, we need to eliminate b,c from the left hand side
Assuimng a⋅b⋅c⋅d≠0,
As ba=cb=dc, we have b2=ca,c2=bd,ad=bc
(b−c)2+(c−a)2+(d−b)
2 =b2+c2−2bc+c2+a2−2ca+d2−2bd+b2
=a2+d2+2(b2+c2−bc−ca−bd)
=a2+d2+2(ca+bd−ad−ca−bd)
=a2+d2−2ad=(a−d)2
Assuimng a⋅b⋅c⋅d≠0,
As ba=cb=dc, we have b2=ca,c2=bd,ad=bc
(b−c)2+(c−a)2+(d−b)
2 =b2+c2−2bc+c2+a2−2ca+d2−2bd+b2
=a2+d2+2(b2+c2−bc−ca−bd)
=a2+d2+2(ca+bd−ad−ca−bd)
=a2+d2−2ad=(a−d)2
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