If a, b, c are in A.P., then show that: () a (b + c), b2 (c + a), c2 (a + b) are also in A.P.

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Given that, вв ( a, b, c ) are in ( A P ) ( Leftrightarrow(b-a)=(-b) ) 1 i) ( a^{2}(b+c), b^{2}(c+a), c^{2}(a+b) ) dets subtract fist and middle term and eastand middle tom ( [a] quad b^{2}(c+a)-a^{2}(b+c) ) [ begin{array}{l} =b^{2} c+a b^{2}-a^{2} b^{2}-a^{2} c =a b(b-a)+cleft(b^{2}-a^{2}right) =a b(b-a)+c(b-a)(b+a) =(b-a)[a b+a c+b c]- end{array} ] ( [b] quad c^{2}(a+b)-b^{2}(c+a) ) [ begin{array}{l} =c^{2} a+c^{2} b-b^{2} c-b^{2} a =left(a-b^{2}+b^{2}right)+b c(c-b) =a(c-b)(c+b)+b c(c-b) =(c-b)[a c+a b+b c]- end{array} ] 7 we equate (a) and(b) we get eq ( ^{n}(1) ) ( (b-a)=(c-b)- ) we got eq ( ^{n}(1) ) thence, the given terms are in ( A P ). to

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