Given a,b,c,d are in G.P then check whether $(a^{2} + b^{2} + c^{2}) (b^{2} + c^{2} + d^{2}) = (a b + b c + c d + d a)^{2}$ .

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Solution

Since $a, b, c, d$ are in G.P.,
we take $b$ to be $a r$ , $c$ to be $a r^{2}, d$ as $a r^{3} .$
Thus, L.H.S. $= (a^{2} + a^{2} r^{2} + a^{2} r^{4}) (a^{2} r^{2} + a^{2} r^{4} + a^{2} r^{6}) = a^{4} r^{2} (1 + r^{2} + r^{4})^{2}$

R.H.S. $= (a^{2} r + a^{2} r^{3} + a^{2} r^{5} + a^{2} r^{3})^{2} = a^{4} r^{2} (1 + 2 r^{2} + r^{4})^{2}$
Hence, clearly the L.H.S. & R.H.S. are not the same.

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