Prove that `Sigmaunderset(r=0)overset(n) 3^r "^(n)C_r = 4^n`

Prove that underset (r=0) overset (n) sum 3^r ""^nC_r =4^n .

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.

Evaluate : underset (r=1) overset (n) sum ""^nC_r 2^r .

Prove that: sum_(r=0)^(n) 3^( r) ""^(n)C_(r) = 4^(n) .

Prove that sum_(r=0)^n 3^r "^nC_r= 4^n .

Prove by mathematical induction that sum_(r=0)^(n)r^(n)C_(r)=n.2^(n-1), forall n in N .

Equation x^(n)-1=0,ngt1,ninN, has roots 1,a_(1),a_(2),...,a_(n),. The value of underset(r=2)overset(n)sum(1)/(2-a_(r)), is

Equation x^(n)-1=0,ngt1,ninN, has roots 1,a_(1),a_(2),...,a_(n),. The value of underset(r=2)overset(n)sum(1)/(2-a_(r)), is

Prove that "^nC_r+2 ^(n)C_(r-1)+ ^(n)C_(r-2) = ^(n+2)C_r .

The vaule of sum_(r=0)^(n-1) (""^(C_(r))/(""^(n)C_(r) + ""^(n)C_(r +1))) is equal to