Prove that `sum_(r=1)^(m-1)(2r^2-r(m-2)+1)/((m-r)^m C_r)=-1/mdot`

Prove that sum_(r=1)^(n)(-1)^(r-1)(1+(1)/(2)+(1)/(3)+...+(1)/(r))^(n)C_(r)=(1)/(n)

m sum_(r=1)^(n)(1)/(n)sqrt((n+r)/(n-r))

Show that the HM of (2n+1)C_(-) and (2n+1)C_(-)(r+1) is (2n+1)/(n+1) xx of (2n)C_(r) Also show that sum_(r=1)^(2n-1)(-1)^(r-1)*(r)/(2nC_(r))=(n)/(n+1)

Show that sum_(k=m)^n ^kC_r=^(n+1)C_(r+1)-^mC_(r+1)

sum_(r=1)^(n)(1)/((r+1)(r+2))*^(n+3)C_(r)=

The value of sum_(r=1)^(3m)(-3)^(r-1).6mC_(2r-1) is

Prove that : (r_1)/(b c)+(r_2)/(c a)+(r_3)/(a b)=1/r-1/(2R)

sum_(r=1)^(n)r^(2)-sum_(m=1)^(n)sum_(r=1)^(m)r is equal to

If k and n are positive integers and s_(k)=1^(k)+2^(k)+3^(k)+...+n^(k), then prove that sum_(r=1)^(m)m+1C_(r)s_(r)=(n+1)^(m+1)-(n+1)