Prove that `sum_(k=1)^(n-1)(n-k)cos(2kpi)/n=-n/2,w h e r engeq3i sa nin t ege r`

Prove that sum_(k=1)^(n-1)(n-k)(cos(2k pi))/(n)=-(n)/(2) wheren >=3 is an 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 sum_(r=1)^(k)(-3)^(r-1)3nC_(2r-1)=0, where k=(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_(k=1)^(n)k2^(-k)=2[1-2^(-n)-n*2^(-(n+1)))

If n be integer gt1, then prove that sum_(r=1)^(n-1) cos (2rpi)/n=-1

Prove that sum_(k=1)^(n-1) ""^(n)C_(k)[cos k x. cos (n+k)x+sin(n-k)x.sin(2n-k)x]=(2^(n)-2)cos nx .

Prove that sum_(r=1)^(n)((1)/(cos theta+cos(2r+1)theta))=(sin n theta)/(2sin theta*cos theta*cos(n+1)theta),(wherenin N)