If `x_(r)=cos theta_(r)+i sin theta_(r),(r=1,2,3,...,n)` and if `x_(1)+x_(2)+...+x_(n)=0` then show that `x_(1)^(-1)+x_(2)^(-1)+...+x_(n)^(-1)=0`

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r), then prove that C_(1)+2c_(2)+3C_(1)+...+nC_(n)=n2^(n-1)...

If Sigma_(r=1)^(n) cos^(-1)x_(r)=0, then Sigma_(r=1)^(n) x_(r) equals

If Sigma_(r=1)^(n) cos^(-1)x_(r)=0, then Sigma_(r=1)^(n) x_(r) equals

If x_(1),x_(2),x_(3),......x2_(n) are in A.P, then sum_(r=1)^(2n)(-1)^(r+1)x_(r)^(2) is equal to

If sum_(r=1)^(n)cos^(-1)x_(r)=0, then sum_(r=1)^(n)x_(r) equals to

(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + ...+ C_(n) x^(n) , show that sum_(r=0)^(n) C_(r)^(3) is equal to the coefficient of x^(n) y^(n) in the expansion of {(1+ x)(1 + y) (x + y)}^(n) .

(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + ...+ C_(n) x^(n) , show that sum_(r=0)^(n) C_(r)^(3) is equal to the coefficient of x^(n) y^(n) in the expansion of {(1+ x)(1 + y) (x + y)}^(n) .

If (1+x+x^(2))^(n) = sum_(r=0)^(2 n) a_(r). x^(r) then a_(1)-2 a_(2)+3 a_(3)-..-2 n. a_(2n) =

If sum_(r=1)^(n)Cos^(-1)x_(r)=0," then "sum_(r=1)^(n)x_(r) equals to