` a, b, c, d` are four different real numbers which are in AP.If `2(a - b) +x (b-c)^2 + (c-a)^3 = 2 (a-d) + (b - d)^2 + (c-d)^3`, then

a,b,c,d are four different real numbers which are in APIf 2(a-b)+x(b-c)^(2)+(c-a)^(3)=2(a-d)+(b-d)^(2)+(c-d)^(3), then

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

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

If a, b, c, d are in A.P. and a, b, c, d are in G.P., show that a^(2) - d^(2) = 3(b^(2) - ad) .

If a, b, c, d and p are different real numbers such that (a^2+b^2+c^2).p^2-2(ab+b c+cd) p +(b^2+c^2+d^2) le 0 , then show that a, b, c and d are in G.P .

If a ,\ b ,\ c ,\ d\ a n d\ p are different real numbers such that: (a^2+b^2+c^2)p^(2)-2(a b+b c+c d)p+(b^2+c^2+d^2)lt=0 , then show that a ,\ b ,\ c and d are in G.P.

Let a ,b ,c ,d be four distinct real numbers in A.P. Then half of the smallest positive valueof k satisfying a(a-b)+k(b-c)^2=(c-a)^3=2(a-x)+(b-d)^2+(c-d)^3 is __________.

If a, b, c, d and p are different real numbers such that (a^2+b^2+c^2)p^2-2(a b+b c+c d)p+(b^2+c^2+d^2)lt=0 , then show that a, b, c and d are in G.P.

If a, b, c, d are four non - zero real numbers such that (d+a-b)^(2)+(d+b-c)^(2)=0 and roots of the equation a(b-c)x^(2)+b(c-a)x+c(a-b)=0 are real equal, then