Evaluate `Sigmaunderset(r=0)overset(n) "^(n)C_(r)3^(r)`

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

Prove that underset (r=0) overset (n) sum 3^r ""^nC_r =4^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 underset(r=1)overset(k)sum(-3)^(r-1) .^(3n)C_(2r-1) = 0 , where k = (3n)/2 and n is an even integer.

Statement-1: sum_(r =0)^(n) (r +1)""^(n)C_(r) = (n +2) 2^(n-1) Statement -2: sum_(r =0)^(n) (r+1) ""^(n)C_(r) x^(r) = (1 + x)^(n) + nx (1 + x)^(n-1)

the value of overset(n)underset(r=2)(Simga)(-2)^(r ) |{:( .^(n-2)C_(r-2),,.^(n-2)C_(r-1),,.^(n-2)C_(r)),(-3,,1 ,,1),(2,,-1,,0):}| (n gt 2)

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

If a_(n) = sum_(r=0)^(n) (1)/(""^(n)C_(r)) , find the value of sum_(r=0)^(n) (r)/(""^(n)C_(r))

Find the sum sum_(r=0)^n^(n+r)C_r .