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.

Answer 1

\[\text { Since a, b, c are in A . P . , we have: } \]

\[2b = a + c\]

\[\text { We have to prove the following: } \]

\[2 b^2 (a + c) = a^2 (b + c) + c^2 (a + b)\]

\[\text { LHS: } 2 b^2 \times 2b (\text{ Given })\]

\[ = 4 b^3 \]

\[\text { RHS: } a^2 b + a^2 c + a c^2 + c^2 b\]

\[ = ac(a + c) + b( a^2 + c^2 )\]

\[ = ac(a + c) + b[(a + c )^2 - 2ac]\]

\[ = ac(2b) + b\left[ \left( 2b \right)^2 - 2ac \right]\]

\[ = 2abc + 4 b^3 - 2abc\]

\[ = 4 b^3 \]

\[\text { LHS = RHS } \]

\[\text { Hence, proved }. \]