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Question
If a, b, c are in G.P., prove that the following is also in G.P.:
a2 + b2, ab + bc, b2 + c2
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Solution
a, b and c are in G.P.
∴ \[b^2 = ac . . . . . . . (1)\]
\[\left( ab + bc \right)^2 = \left( ab \right)^2 + 2a b^2 c + \left( bc \right)^2 \]
\[ \Rightarrow \left( ab + bc \right)^2 = \left( ab \right)^2 + a b^2 c + a b^2 c + \left( bc \right)^2 \]
\[ \Rightarrow \left( ab + bc \right)^2 = a^2 b^2 + ac\left( ac \right) + b^2 \left( b^2 \right) + b^2 c^2 \left[ \text { Using } (1) \right]\]
\[ \Rightarrow \left( ab + bc \right)^2 = a^2 \left( b^2 + c^2 \right) + b^2 \left( b^2 + c^2 \right)\]
\[ \Rightarrow \left( ab + bc \right)^2 = \left( b^2 + c^2 \right)\left( a^2 + b^2 \right)\]
\[\text { Therefore }, \left( a^2 + b^2 \right), \left( b^2 + c^2 \right) \text { and }\left( ab + bc \right) \text { are also in G . P } . \]
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