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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)
\[\left( b^2 + c^2 \right)^2 = \left( b^2 \right)^2 + 2 b^2 c^2 + \left( c^2 \right)^2 \]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = \left( ac \right)^2 + b^2 c^2 + b^2 c^2 + \left( bd \right)^2 \left[\text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = a^2 c^2 + a^2 d^2 + b^2 c^2 + b^2 d^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = a^2 \left( c^2 + d^2 \right) + b^2 \left( c^2 + d^2 \right)\]
\[ \Rightarrow \left( b^2 + c^2 \right)^2 = \left( a^2 + b^2 \right)\left( c^2 + d^2 \right)\]
\[\text {Therefore, } \left( a^2 + b^2 \right), \left( c^2 + d^2 \right)\text{ and } \left( b^2 + c^2 \right) \text { are also in G . P } .\]
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