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

[sum_(r=1)^(n)((1)/(r)-(1)/(r+1))]

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)

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

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 ^nC_(r)+^(n-1)C_(r)+...+^(r)C_(r)=^(n+1)C_(r+1)

sum_(r=1)^n r (n-r +1) is equal to :

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