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Question
If a, b, c is in A.P., then show that:
a2 (b + c), b2 (c + a), c2 (a + b) are also in A.P.
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Solution
\[\text { Since a, b, c are in A . P . , we have: } \]
\[2b = a + c\]
\[\text { We have to prove the following: } \]
\[2 b^2 (a + c) = a^2 (b + c) + c^2 (a + b)\]
\[\text { LHS: } 2 b^2 \times 2b (\text{ Given })\]
\[ = 4 b^3 \]
\[\text { RHS: } a^2 b + a^2 c + a c^2 + c^2 b\]
\[ = ac(a + c) + b( a^2 + c^2 )\]
\[ = ac(a + c) + b[(a + c )^2 - 2ac]\]
\[ = ac(2b) + b\left[ \left( 2b \right)^2 - 2ac \right]\]
\[ = 2abc + 4 b^3 - 2abc\]
\[ = 4 b^3 \]
\[\text { LHS = RHS } \]
\[\text { Hence, proved }. \]
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