Prove that; $a^{2} - b^{2}, b^{2} - c^{2}, c^{2} - d^{2}$ are in $G . P .$

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Solution

We have to prove that
$(a^{2} - b^{2}) (c^{2} - d^{2}) = (b^{2} - c^{2})^{2}$
$L . H . S . = a^{2} c^{2} + b^{2} d^{2} - (b^{2} c^{2} + a^{2} d^{2})$
$= b^{4} + c^{4} - 2 b^{2} c^{2} = (b^{2} - c^{2})^{2}$ , by $(1)$ of $(a)$
Therefore, $a^{2} - b^{2}, b^{2} - c^{2}, c^{2} - d^{2}$ are in $G . P .$

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