Prove that `sum_(r = 1)^n r^3 ((n_C_r)/(C_(r - 1)))^2 = (n (n + 1)^2 (n+2))/(12)`

Prove that sum_(r = 0)^n r^2 . C_r = n (n +1).2^(n-2)

Prove that sum_(r = 1)^(n+1) (2^(r +1) C_(r - 1) )/(r (r + 1)) = (3^(n+2) - 2n - 5)/((n+1)(n+2))

Prove that sum_(r = 0)^n r^3 . C_r = n^2 (n +3).2^(n-3)

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

Prove that sum_(r=0)^(2n)(r. ^(2n)C_r)^2=n^(4n)C_(2n) .

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

Find the sum of sum_(r=1)^n(r^n C_r)/(^n C_(r-1) .

If n is a positive integer, prove that underset(r=1)overset(n)sumr^(3)((""^(n)C_(r))/(""^(n)C_(r-1)))^(2)=((n)(n+1)^(2)(n+2))/(12)

Prove that sum_(r=0)^ssum_(s=1)^n^n C_s^n C_r=3^n-1.

Prove that sum_(r=0)^(n) ""^(n)C_(r).(n-r)cos((2rpi)/(n)) = - n.2^(n-1).cos^(n)'(pi)/(n) .