InterviewSolution
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InterviewSolution
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If `a,b,c` are in AP, than show that `a^(2)(b+c)+b^(2)(c+a)+c^(2)(a+b)=(2)/(9)(a+b+c)^(3)`. |
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Answer» `therefore a,b,c` are in AP. `therefore b=(a+c)/(2)i.e., 2b=a+c " " "…….(i)"` LHS`=a^(2)(b+c)+b^(2)(c+a)+c^(2)(a+b)` `=(a^(2)b+a^(2)c)+b^(2)(2b)+(c^(2)a+c^(2)b)` `=b(a^(2)+c^(2))+ac(a+c)+2b^(3)` `=b[(a+c)^(2)-2ac]+ac(2b)+2b^(3)` `=b(a+c)^(2)+2b^(3)=b(2b)^(2)+2b^(3)=6b^(3)` RHS `=(2)/(9)(a+b+c)^(3)=(2)/(9)(2b+b)^(3)` `=(2)/(9)xx27b^(3)=6b^(3)` Hence, LHS=RHS. |
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