`I fn >2t h a nsum_(r=0)^n(-1)^r(n-r)(n-r+1)C_r=(A)0(B)n(C)2^n(D)(n-1)2^(n-1)`

If (1+x)^n=sum_(r=0)^n C_rx^r then prove that sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+1))

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .

If sum_(r=0)^n(2r+3)/(r+1), C_r=((n+k)*2^(n+1)-1)/(n+1) then 'k' is

Deduce that: sum_(r=0)^(n)*^(n)C_(r)(-1)^(n)(1)/((r+1)(r+2))=(1)/(n+2)

Let for n in N, f(n)=sum_(r=0)^(n)(-1)^(r)(C_(r)2^(r+1))/((r+1)(r+2))

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 -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)}