Prove that `(1 + i)^(n) + (1 - i)^(n) = 2^((n + 2)/(2)) cos (n pi)/(4)`

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

i^(n) + i^(n+1) + i^(n + 2) + i^(n + 3)

Prove that ((n + 1)/(2))^(n) gt n!

Prove that n! ( n +2) = n ! + ( n + 1) !

Prove that : (1 + i)^(4n) and (1 + i)^(4n + 2) are real and purely imaginary respectively.

If n_1, n_2 are positive integers, then (1 + i)^(n_1) + ( 1 + i^3)^(n_1) + (1 + i_5)^(n_2) + (1 + i^7)^(n_2) is real if and only if :

Prove that .^(2n)C_(n) = ( 2^(n) xx 1 xx 3 xx …(2n-1))/(n!)

Prove that ((2n)!)/(n!) =2^(n) (1,3,5,……..(2n-1)) .