Question

the least positive integer n such that (2i/1+ i )^n is positive integer is (1) I6 (2)8 (3) 4 (4) 2​

Answer 1

Step-by-step explanation:

Given  

the least positive integer n such that (2i/1+ i )^n is positive integer is

• So (2i / 1 + i)^n is positive.
• Rationalizing the denominator we get
• (2i / 1 + i x 1 – i / 1 – i)^n
• (2i (1 – i) / (1 – i^2)^n
•  (2i (1 – i) / 1 – (- 1))^n
•  (2i (1 – i) / 2)^n
•  (1 – i^2)^n
•    (i – i^2) ^n
•    (i + 1)^n
• Now if n = 2
•  (i – i^2)^2 = 1 + i^2 + 2i
•                    = 2i
• So n = 2 is not satisfied.
• Now if n = 3 we get
•    (i – i^2)^3 = (1 + i)^2(1 + i)
•                       = 2i (1 + i)
•                       = 2i – 2
• So n= 3 is not satisfied.
• Now if n = 4 is not satisfied as it gives negative value.
• Now if n = 8 we get
• So (i – i^2)^8
•   ((i – i^2)^4)^2
•  (- 4)^2
•     = 16  
• Therefore n = 8 is satisfied.

Answer 2

Answer:

n=8 is the right answer ........ . ....