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"If "(1)/(a+b),(1)/(2b),(1)/(b+c) are in...

`"If "(1)/(a+b),(1)/(2b),(1)/(b+c)` are in A.P., then prove that a, b, c are in G.P.

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To prove that \( a, b, c \) are in Geometric Progression (G.P.) given that \( \frac{1}{a+b}, \frac{1}{2b}, \frac{1}{b+c} \) are in Arithmetic Progression (A.P.), we will follow these steps: ### Step 1: Set up the condition for A.P. Since the terms \( \frac{1}{a+b}, \frac{1}{2b}, \frac{1}{b+c} \) are in A.P., we can use the property of A.P. which states that the middle term is the average of the other two terms. Therefore, we can write: \[ \frac{1}{2b} = \frac{\frac{1}{a+b} + \frac{1}{b+c}}{2} \] ...
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