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Question
If a, b, c, d are in G.P., prove that:
(a2 − b2), (b2 − c2), (c2 − d2) are in G.P.
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Solution
a, b, c and d are in G.P.
\[\therefore b^2 = ac\]
\[ad = bc \]
\[ c^2 = bd\] .......(1)
\[\left( b^2 - c^2 \right)^2 = \left( b^2 \right)^2 - 2 b^2 c^2 + \left( c^2 \right)^2 \]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = \left( ac \right)^2 - b^2 c^2 - b^2 c^2 + \left( bd \right)^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = a^2 c^2 - b^2 c^2 - a^2 d^2 + b^2 d^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = c^2 \left( a^2 - b^2 \right) - d^2 \left( a^2 - b^2 \right)\]
\[ \Rightarrow \left( b^2 - c^2 \right)^2 = \left( a^2 - b^2 \right)\left( c^2 - d^2 \right)\]
\[\text { Therefore, } \left( a^2 - b^2 \right), \left( b^2 - c^2 \right) \text { and } \left( c^2 - d^2 \right) \text { are also in G . P } .\]
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