Question

Find the smallest positive integer value of n for which (1+i)^n/(1-i)^n-2 is a real number

Answer 1

we have to find n in such a way that $\frac{(1+i)^n}{(1-i)^{n-2}}$ is a real number

e.g., (1 + i)^n/(1 - i)^(n-2)

= (1 + i)^n(1 + i)^(n-2)/(1 - i)^(n-2).(1+i)^(n-2)

= (1 + i)^(n+n-2)/{1² - (i)²}^(n-2)

=(1 + i)^(2n-2)/(1 + 1)^(n-2)

=(1 + i)^(2n-2)/2^(n-2)

=(1² + i² + 2i)^(n-1)/2^(n-2)

=(2i)^(n-1)/2^(n-2)

=2^(n-1)i^(n-1)/2^(n-2)

= 2i^(n-1) it will be real only when n = 1

Answer 2

I hope it will help you!!!