Prove that `(sqrt(2)-1)^(2n)`

Prove that ((-1+sqrt(-3))/(2))^(n)+((-1-sqrt(-3))/(2))^(n) =2, when n is positive integer multipel of 3, =-1 when n is positive integer but not a multiple of 3.

Prove that ((2n)!)/(2^(2n)(n!)^(2))<=(1)/(sqrt(3n+1)) for all n in N

Prove that 2^(n)>1+n sqrt(2^(n-1)),AA n>2 where n is a positive integer.

Prove that 1+(1)/(sqrt2)+(1)/(sqrt3)+....+(1)/(sqrtn) ge sqrtn, AA n in N

Prove that 1+(1)/(sqrt2)+(1)/(sqrt3)+....+(1)/(sqrtn) ge sqrtn, AA n in N

Prove that : (1)/(sqrt(2)+1)+ (1)/(sqrt(3)+sqrt(2))+ (1)/(2+sqrt(3))=1

If sqrt(1-x^(2n))+sqrt(1-y^(2n))=a^(n)(x^(n)-y^(n)) prove that y^(n-1)*sqrt(1-x^(2n))dy=x^(n-1)sqrt(1-y^(2n))dx

If a_(1),a_(2),a_(3)... are in A.P then prove that (1)/(sqrt(a)_(1)+sqrt(a)_(2))(+)/(sqrt(a)_(2)+sqrt(a)_(3))+...+(1)/(sqrt(a)_(n-1)+sqrt(a)_(n))=(n-1)/(sqrt(a)_(n)+sqrt(a)_(1))

If C_r=(n !)/([r !(n-r)]), the prove that sqrt(C_1)+sqrt(C_2)+.......sqrt(C_n) lt sqrt(n(2^n-1)) ="">

If C_r=(n !)/([r !(n-r)]), the prove that sqrt(C_1)+sqrt(C_2)+.......sqrt(C_n) lt sqrt(n(2^n-1)) ="">