`(C_(r)+C_(r-1))/(C_(r-1))=(n+1)/(r)`

Prove that : (""^(n)C_(r+1))/(""^(n)C_(r))=(n-r)/(r+1)

Let n and r be no negative integers suych that r<=n. Then,^(n)C_(r)+^(n)C_(r-1)=^(n+1)C_(r)

Prove that .^(n)C_(r)+^(n)C_(r-1)=^(n+1)C_(r)

Property: ( i )nC_(r)=nC_(n-r)( ii) n(C_(r))/(r+1)=(n+1)(C_(r+1))/(n+-1)

f(n)=sum_(r=1)^(n)[r^(2)(*^(n)C_(r)-^(n)C_(r-1))+(2r+1)^(n)C_(r)] then f(30) is

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)

Statement 1:^mC_(r)+mC_(r-1)^(n)C_(1)+mC_(r-2)^(n)C_(2)+...+^(n)C_(r)=0, if m+n

the roots of the equations |{:(.^(x)C_(r),,.^(n-1)C_(r),,.^(n-1)C_(r-1)),(.^(x+1)C_(r),,.^(n)C_(r),,.^(n)C_(r-1)),(.^(x+2)C_(r),,.^(n+1)C_(r),,.^(n+1)C_(r-1)):}|=0

C_(0)-C_(1)+C_(2)-C_(3)+......+(-1)^(r)C_(r)=((-1)^(r)(n-1)!)/(r!*(n-r-1)!)

Prove that (r+1)^(n)C_(r)-r^(n)C_(r)+(r-1)^(n)C_(2)-^(n)C_(3)+...+(-1)^(r)n_(C_(r))=(-1)^(r_(n-2))C_(r)