if a,b,c are in ap then a^2(b+c)+b^2(a+c)+c^2(a+b)=----------

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Dear Student,
Please find below the solution to your problem.

if a2(b+c) , b2(a+c), c2(a+b) are in AP then

2b2(a+c) = a2(b+c) + c2(a+b) (we have to prove this) ..............................1
RHS:
= a2 (b+c) + c2 (a+b)
= a2b +a2c +c2a +c2b
=b(a2 +c2) + ac(a+c)
since a,b,c are in AP so b=a+c/2
RHS = (a+c)(a2 +b2)/2 + ac(a+c)
=(a+c)(a2 +b2 +2ac)/2
=(a+c)3 /2 or 2(a+c) b2 (after putting a+c =b)
hence proved

Thanks and Regards

Dear Student,
Please find below the solution to your problem.

if a2(b+c) , b2(a+c), c2(a+b) are in AP then

2b2(a+c) = a2(b+c) + c2(a+b) (we have to prove this) ..............................1
RHS:
= a2 (b+c) + c2 (a+b)
= a2b +a2c +c2a +c2b
=b(a2 +c2) + ac(a+c)
since a,b,c are in AP so b=a+c/2
RHS = (a+c)(a2 +b2)/2 + ac(a+c)
=(a+c)(a2 +b2 +2ac)/2
=(a+c)3 /2 or 2(a+c) b2 (after putting a+c =b)
hence proved

Thanks and Regards