`sum_(r=0)^n(-1)^r^n C_r[1/(2^r)+3/(2^(2r))+7/(2^(3r))+(15)/(2^(4r))+ u ptomt e r m s]= (2^(m n)-1)/(2^(m n)(2^n-1))`

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

underset(r=0)overset(n)(sum)(-1)^(r).^(n)C_(r)[(1)/(2^(r))+(3^(r))/(2^(2r))+(7^(r))/(2^(3r))+(15^(r))/(2^(4r))+ . . .m" terms"]=

If sum_(r=0)^(n) (r)/(""^(n)C_(r))= sum_(r=0)^(n) (n^(2)-3n+3)/(2.""^(n)C_(r)) , then

If sum_(r=0)^n((r^3+2r^2+3r+2)/(r+1)) ^nC_r=(2^4+2^3+2^2-2)/3

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

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

f(n)=sum_(r=1)^(n) [r^(2)(""^(n)C_(r)-""^(n)C_(r-1))+(2r+1)(""^(n)C_(r ))] , then

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

Prove that (3!)/(2(n+3))=sum_(r=0)^n(-1)^r((^n C_r)/(^(r+3)C_r))