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Question
If \[\frac{1}{a + b}, \frac{1}{2b}, \frac{1}{b + c}\] are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.
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Solution
Here,
\[\frac{1}{a + b}, \frac{1}{2b} \text { and } \frac{1}{b + c} \text { are in A . P } . \]
\[\therefore 2 \times \frac{1}{2b} = \frac{1}{a + b} + \frac{1}{b + c}\]
\[ \Rightarrow \frac{1}{b} = \frac{b + c + a + b}{\left( a + b \right)\left( b + c \right)}\]
\[ \Rightarrow \left( a + b \right)\left( b + c \right) = b\left( 2b + a + c \right)\]
\[ \Rightarrow ab + ac + b^2 + bc = 2 b^2 + ab + bc\]
\[ \Rightarrow 2 b^2 - b^2 = ac\]
\[ \Rightarrow b^2 = ac\]
\[\text { Thus, a, b and c are in G . P } .\]
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