Evaluate: `sum_(r=1)^n(3^r-2^r)`

Evaluate sum_(r=1)^(n-1) cos^(2) ((rpi)/(n)) .

If f(n)=sum_(r=1)^(n) r^(4) , then the value of sum_(r=1)^(n) r(n-r)^(3) is equal to

Evaluate S=sum_(r=0)^(n-1)(1)/(sqrt(4n^(2)-r^(2)))as n rarrinfty .

If n in N such that is not a multiple of 3 and (1+x+x^(2))^(n) = sum_(r=0)^(2n) a_(r ). X^(r ) , then find the value of sum_(r=0)^(n) (-1)^(r ).a_(r).""^(n)C_(r ) .

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

If D_(r) = |(r,1,(n(n +1))/(2)),(2r -1,4,n^(2)),(2^(r -1),5,2^(n) -1)| , then the value of sum_(r=1)^(n) D_(r) , is

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

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

If n is a positive integer and C_r = ""^nC_r then find the value of sum_(x = 1)^n r^2 ((C_r)/(C_(r - 1)) )