If a,b,c,d are in G.P.then prove that $(a^{2} + a c + c^{2}) (b^{2} + b d + d^{2}) = {(a b + b c + c d)}^{2}$ .

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Solution

Since $a, b, c, d$ are in $G . P .$

$∴ b = a r, c = a r^{2}, d = a r^{3}$

$L . H . S . = a^{2} (1 + r^{2} + r^{4}) a^{2} (r^{2} + r^{4} + r^{6})$

$L . H . S . = a^{4} r^{2} (1 + r^{2} + r^{4})^{2}$

$R . H . S . = [a^{2} (r + r^{3} + r^{5})]^{2}$

$R . H . S . = a^{4} r^{2} (1 + r^{2} + r^{4})^{2} = L . H . S .$

Hence Proved.

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If a, b, c, d are in GP, prove that

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