Prove that `sum_(1lt=ilt) sum_(jlt=n)(i) = (n (n^2 - 1))/(6)`

Prove that sum_(0lt=i)sum_(ltjlt=n) (C_i +C_j)= n.2^n

If a_(i) gt 0 (i = 1,2,3,….n) prove that sum_(1 le i le j le n) sqrt(a_(i)a_(j)) le (n - 1)/(2) (a_(1) + a_(2) + …. + a_(n))

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

sum_(n=1)^(oo) (2n)/(n!)=

If (1 + x)^(n) = sum_(r=0)^(n) C_(r),x^(r) , then prove that (sumsum)_(0leiltjlen) ((i)/(C_(i)) + (j)/(C_(j))) = (n^(2))/(2) sum_(r=0)^(n) (1)/(C_(r)) .

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_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

sum_(n=1)^(oo)(2 n^2+n+1)/((n!)) =

sum_(n=1) ^(oo) (1)/((2n-1)!)=