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Question
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are in A.P.
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Solution
\[\text { Since a, b, c are in A . P . , we have: } \]
\[2b = a + c\]
\[\text { We have to prove the following }: \]
\[2(c + a - b) = (b + c - a) + (a + b - c)\]
\[\text { LHS: }2(c + a - b)\]
\[ = 2(2b - b) \left( \because 2b = a + c \right)\]
\[ = 2b\]
\[\text { RHS }: (b + c - a) + (a + b - c)\]
\[ = 2b\]
\[\text { LHS = RHS }\]
\[\text { Hence, proved } .\]
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