Prove that `sum_(k=1)^(n-1)(n-k)cos(2kpi)/n=-n/2`, where`ngeq3` is an integer

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_(k=1)^(n)k2^(-k)=2[1-2^(-n)-n*2^(-(n+1)))

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.

sum_(r=1)^k(−3)^(r−1) .^(3n)C_(2r−1)=0, where k=(3n)/2 and n is an even integer

Prove that (a) (1+i)^n+(1-i)^n=2^((n+2)/2).cos((npi)/4) , where n is a positive integer. (b) (1+isqrt(3))^n+(1-isqrt(3)^n=2^(n+1)cos((npi)/3) , where n is a positive integer

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 , where k = 3n//2 and n is an even integer.

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