Prove that `sum_(r=1)^k (-3)^(r-1)` `"^(3n)C_(2r-1) =0 `, where k=(3n)/2 and n is an even integer

Prove that sum_(r=1)^k(-3)^(r-1)^(3n)C_(2r-1)=0,w h e r ek=3n//2 and n is an even integer.

Prove that sum_(r=1)^(k)(-3)^(r-1)C(3n,2r-1)=0 where k=(3n)/(2) and n is an even positive integer.

Prove that sum_(r=1)^(k)(-3)^(r-1)3nC_(2r-1)=0, where k=3n/2 and n is an even integer.

Prove that underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C_(2r-1) = 0 , where k = (3n)/2 and n is an even integer.

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

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

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

The value of sum_(r=0)^(3n-1)(-1)^r ^(6n)C_(2r+1)3^r is

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

The value of sum_(r=0)^(3n-1)(-1)^r .^(6n)C_(2r+1)3^r is