R D Sharma|Statement Sums

Statement-1 sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = nx (1 - x)^(n -1) Statement-2: sum_(r=0)^(n)r ""^(n)C_(r) x^(r) (-1)^(r) =0

Statement:1: If sum _(r=1)^(n) sin(x_(r)) = n , then sum_(n=1)^(n) cot (x_(r)) =n . And Statement-2: The number of solutions of the equation cosx =x is 1.

Statement -2: sum_(r=0)^(n) (-1)^( r) (""^(n)C_(r))/(r+1) = (1)/(n+1) Statement-2: sum_(r=0)^(n) (-1)^(r) (""^(n)C_(r))/(r+1) x^(r) = (1)/((n+1)x) { 1 - (1 - x)^(n+1)}

Statement-1: sum_(r =0)^(n) (r +1)""^(n)C_(r) = (n +2) 2^(n-1) Statement -2: sum_(r =0)^(n) (r+1) ""^(n)C_(r) x^(r) = (1 + x)^(n) + nx (1 + x)^(n-1)

Statement 1: sum sum_(0le ilt j le n)(i/ (^n c_i)+j/(^nc_j)) is equal to(n^2)/2a , where a ,sum_(r="0)^(n) 1/(^n"" c_r)="" .="" statement 2:sum_(r=0)^(n) r/(^n" c_r)="sum_(r=0)^(n)(n-r)/(^n" .

Statement-1: sum_(r=0)^(n) (1)/(r+1) ""^(n)C_(r) = (1)/((n+1)x) {( 1 + x)^(n+1) -1}^(-1) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))/(r+1) = (2^(n+1))/(n+1) .

Statement -1: sum_(r=0)^(n) r(""^(n)C_(r))^(2) = n (""^(2n -1)C_(n-1)) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))^(2)= ""^(2n)C_(n)

Statement 1:sum_(0<=i<=j)sum_(<=n)C(n,i)C(n,j)=2^(2n)-C(2n,n) Statement 2:(sum_(r=0)^(n))C(n,r))^(2)=sum_(r=0)^(n)(C(n,r))^(2)+2sum_(0<=i<=j)sum_(<=n)C(n,i)C(n,j)