The positive real numbers a, b, c satisfy: aligned & a 2 b+1 + 2 b 3 c+1 + 3 c a+1 =1 & 1 a+1 + 1 2 b+1 + 1 3 c+1 =2 aligned What is the value of 1 a + 1 b + 1 c ?

Step-by-step solution

3 c a+1 + a 2 b+1 + 2 b 3 c+1 =1 ......(i) 1 a+1 + 1 2 b+1 + 1 3 c+1 =2 ......(ii) Add (i) and (ii) 3 c+1 a+1 + a+1 2 b+1 + 2 b+1 3 c+1 =3 aligned & A.M. G.M. & ( 3 c+1 a+1 + a+1 2 b+1 + 2 b+1 3 c+1 3 ) (1)^ 1 3 & 3 c+1 a+1 =1 and 2 b+1 3 c+1 =1 & a+1 2 b+1 =1 & (a+1)=(2 b+1)=(3(+1))=k & 1 k + 1 k + 1 k =2 3 k =2 k= 3 2 & a= 1 2 , b= 1 4 , c= 1 6 & 1 a + 1 b + 1 c =2+4+6=12 aligned