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

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

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

If (i)^(2) = -1, (i)^(2) + (i)^(4) + (i)^(6) + ...... to (2n + 1) terms =

Simplify the following : (i) i^7 " " (ii) i^(1729) " " (iii) i^(-1924) + i^(2018) " " (iv) sum_(n=1)^(102) i^(n) " " (v) i i^2 i^3 …..i^(40)

The value of sum_(k=0)^(n)(i^(k)+i^(k+1)) , where i^(2)= -1 , is equal to

Evaluate the following: (i) i^(135) " "(ii) i^(19) " " (iii) i^(-999) " " (iv) (-sqrt(-1))^(4n + 2), n in N .

Find the value of n if (i) ( n + 1) ! = 20 ( n-1 ) ! ( ii) 1/( 8!) + 1/( 9 !) = n/( 10 !)

The value of sum_(i=1)^(13) (n^(n) + i^(n-1)) is

If b_i=1-a_i na = Sigma_(i=1)^(n) a_i, nb = Sigma_(i=1)^(n) b_i " then " Sigma_(i=1)^(n) a_b_i+Sigma_(i=1)^(n)(a_i-a)^2=