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Question
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
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Solution
Since
\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., we have:
\[\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b}\]
\[ \Rightarrow \frac{\left( a - b \right)}{ab} = \frac{\left( b - c \right)}{bc}\]
\[ \Rightarrow \frac{\left( a - b \right)}{a} = \frac{\left( b - c \right)}{c}\]
\[ \Rightarrow \left( a - b \right)c = a\left( b - c \right)\]
\[ \Rightarrow ac - bc = ab - ac\]
Hence, bc, ca, ab are in A.P.
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