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(A) Given that $a^2(b + c), b^2(c + a), c^2(a + b)$ are in $AP$.Dividing each term by $abc$,we get:$\frac{a^2(b + c)}{abc}, \frac{b^2(c + a)}{abc}, \f

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Arithmetic Progression

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(A) Given that $a^2(b + c), b^2(c + a), c^2(a + b)$ are in $AP$.Dividing each term by $abc$,we get:$\frac{a^2(b + c)}{abc}, \frac{b^2(c + a)}{abc}, \frac{c^2(a + b)}{abc}$ are in $AP$.This simplifies to:$\frac{a(b + c)}{bc}, \frac{b(c + a)}{ac}, \frac{c(a + b)}{ab}$ are in $AP$.$\frac{ab + ac}{bc}, \frac{bc + ab}{ac}, \frac{ca + bc}{ab}$ are in $AP$.Adding $1$ to each term:$\frac{ab + ac}{bc} + 1, \frac{bc + ab}{ac} + 1, \frac{ca + bc}{ab} + 1$ are in $AP$.$\frac{ab + ac + bc}{bc}, \frac{bc + ab + ac}{ac}, \frac{ca + bc + ab}{ab}$ are in $AP$.Dividing by $(ab + bc + ca)$,we get:$\frac{1}{bc}, \frac{1}{ac}, \frac{1}{ab}$ are in $AP$.Multiplying by $abc$,we get:$a, b, c$ are in $AP$.

If $a^2(b + c), b^2(c + a), c^2(a + b)$ are in Arithmetic Progression $(AP)$,then $a, b, c$ are in which progression?

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