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

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

Show that ((19 + 9i)/(5 - 3i))^(15) - ((8 + i)/(1 + 2i))^(15) is purely imaginary

Show that ( 2 + i sqrt(3))^(10) - (2 - i sqrt(3))^(10) is purely imaginary.

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

Consider two complex numbers alphaa n dbeta as alpha=[(a+b i)//(a-b i)]^2+[(a-b i)//(a+b i)]^2 , where a ,b , in R and beta=(z-1)//(z+1), w here |z|=1, then find the correct statement: (a)both alphaa n dbeta are purely real (b)both alphaa n dbeta are purely imaginary (c) alpha is purely real and beta is purely imaginary (d) beta is purely real and alpha is purely imaginary

Reduce (4+3i)/(5-4i) in x+iy form, find its real and imaginary parts.

Write (3 + 4i)/(5 - 12 i) in the x + iy form and hence find real and imaginary parts.

Prove that 4n^(2)-1 is an odd number for all n in N

Show that ((19+9i)/(5-3i))^(15)-((8+i)/(1+2i))^(15) is purely imaginary.