Question

If a, b,c are in A. P show that a^2(b+c), b^2(c+a),c^2(a+b) are also in A. P

Answer 1

1)  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
 
2)      (ab+ac)/bc , (bc+ba)/ac ,(ca+bc)/ab are in AP so
                 2(bc+ba)/ac    =    (ab+ac)/bc + (ac+bc)/ab          we have to prove this
         multiplying the equation by abc
      now we get
                  2b2(a+c) = a2(b+c) + c2(a+b)               we have to prove this
 this expression is same as of eq 1 of previous ans ,so now  take RHS and prove as i have done in previous ans...
Please mark it as brainiest answer .........