Let a , b , c , d be four distinct real numbers in A.P. Find the smallest positive value of k satisfying 2(a-b)+k(b-c)^2+(c-a)^3=2(a-d)+(b-d)^2+(c-d)^3

Step-by-step solution

Since a, b, c, d are in A.P. b - a = c - b = d - c = D (let common difference) aligned & d=a+3 D & a-d=-3 D and d=b+2 D b-d=-2 d aligned Also, c=a+2 D c-a=2 D aligned & Given equation 2(a-b)+k(b-c)^2+(c-a)^3=2(a-d)+(b-d)^2+(c-d)^3 & Becomes -2 D+k D^2+(2 D)^3=-6 D+4 D^2-D^3 & 9 D^2+(k-4) D+4=0 aligned Since D is real (k-4)^2-4(4)(9) 0 aligned & k ^2-8 k -128 0 ( k -16)( k +8) 0 & k (- ,-8] [16, ) aligned Hence smallest positive value of k=16