Let `"Delta"_r=|r x(n(n+1))/2 2r-1y n^2 3r-2z(n(3n-1))/2|dot` Show that `sum_(r=1)^n"Delta"_r=0`

Let Delta_r=|[r , x , (n(n+1))/2] , [2r-1 , y , n^2] , [3r-2 , z , (n(3n-1))/2]|dot Show that sum_(r=1)^n Delta_r=0

Let Delta_r=|[r-1,n,6],[(r-1)^2,2n^2,4n-2],[(r-1)^3,3n^3,3n^2-3n]| . Show that sum_(r=1)^n Delta_r is contant.

Let "T"_r=|[r ,x,(n(n+1))/2], [2r-1,y, n^2],[ 3r-2,z,(n(3n-1))/2]| .where T is the determinant of the given matrix, Then show that sum_(r=1)^n"T" _r=0 .

If "Delta"_r=|[2^(r-1),2. 3^(r-1),4. 5^(r-1)],[x, y ,z],[2^n-1, 3^n-1, 5^n-1]| Show that sum_(r=1)^n"Delta"_r = Constant

If Delta_r=|[2^(r-1) , 2.3^(r-1) , 4. 5^(r-1)] , [x , y , z] , [2^n-1 , 3^n-1 , 5^n-1]|dot Show that sum_(r=1)^n Delta_r=Con s t a n t

If D_(r) = |(r,1,(n(n +1))/(2)),(2r -1,4,n^(2)),(2^(r -1),5,2^(n) -1)| , then the value of sum_(r=1)^(n) D_(r) , is

If Delta_(r) = |(1,r,2^(r)),(2,n,n^(2)),(n,(n(n+1))/(2),2^(n+1))| , then the value of sum_(r=1)^(n) Delta_(r) is

If Delta_(r) = |(2^(r -1),((r +1)!)/((1 + 1//r)),2r),(a,b,c),(2^(n) -1,(n +1)! -1,n(n +1))| , then sum_(r =1)^(n) Delta_(r) is equal to

Prove that sum_(r=0)^n r(n-r)(^nC_ r)^2=n^2(^(2n-2)C_n)dot

If A_r=|(1,r,2^r),(2,n,n^2),(n,(n(n+1))/2 ,2^(n+1))| , then the value of sum_(r=1)^n A_r is n (b) 2n (c) -2n (d) n^2