If a,b,c,d are distinct integers in an A.P. such that `d=a^(2)+b^(2)+c^(2)`, then find the value of a+b+c+d.

If a, b, c, d are distinct integers in A. P. Such that d=a^2+b^2+c^2 , then a + b + c + d is

Let a , b , c , d be positive integers such that (log)_a b=3/2a n d(log)_c d=5/4dot If (a-c)=9, then find the value of (b-d)dot

If a ,b ,c ,d ,e are in A.P., the find the value of a-4b+6c-4d+edot

If a , b , c are consecutive positive integers and (log(1+a c)=2K , then the value of K is logb (b) loga (c) 2 (d) 1

If a, b, c, d are in G.P., then (a + b + c + d)^(2) is equal to

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, d and p are distinct non - zero real numbers such that (a^2+b^2 + c^2) p^2 - 2(ab+bc+cd)p+(b^2 + c^2 +d^2) le 0 then n. Prove that a, b, c, d are in G. P and ad = bc

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

If a ,b ,c ,da n dp are distinct 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 prove that a ,b ,c , d are in G.P.

If A ,B ,C ,D are four distinct point in space such that A B is not perpendicular to C D and satisfies vec A Bdot vec C D=k(| vec A D|^2+| vec B C|^2-| vec A C|^2-| vec B D|^2), then find the value of kdot