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Question
If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:
a (b +c), b (c + a), c (a +b) are in A.P.
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Solution
\[\text { Given }: \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \text { are in A . P } . \]
\[ \therefore \frac{2}{b} = \frac{1}{a} + \frac{1}{c}\]
\[ \Rightarrow 2ac = ab + bc . . . . (1)\]
\[\text { To prove: } a(b + c), b(c + a), c(a + b) \text { are in A . P } . \]
\[ \Rightarrow 2b(c + a) = a(b + c) + c(a + b)\]
\[\text { LHS: } 2b(c + a)\]
\[ = 2bc + 2ba\]
\[\text { RHS: } a(b + c) + c(a + b)\]
\[ = ab + ac + ac + bc\]
\[ = ab + 2ac + bc\]
\[ = ab + ab + bc + bc (\text { From }(1))\]
\[ = 2ab + 2bc\]
\[ \therefore\text { LHS = RHS }\]
\[\text { Hence, proved }.\]
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