For positive integers the value of expression , here is a real number, if and only if
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: D
Solution
Since i³ = -i, i⁵ = i, i⁷ = -i, the expression becomes:
(1+i)n₁ + (1-i)n₁ + (1+i)n₂ + (1-i)n₂
Now (1+i) and (1-i) are complex conjugates, so (1+i)n + (1-i)n is always real for any positive integer n (sum of conjugates = 2·Re[(1+i)n]).
Thus the entire expression = 2·Re[(1+i)n₁] + 2·Re[(1+i)n₂], which is real for ALL positive integers n₁ and n₂, with no restriction needed between them.
(1+i)n₁ + (1-i)n₁ + (1+i)n₂ + (1-i)n₂
Now (1+i) and (1-i) are complex conjugates, so (1+i)n + (1-i)n is always real for any positive integer n (sum of conjugates = 2·Re[(1+i)n]).
Thus the entire expression = 2·Re[(1+i)n₁] + 2·Re[(1+i)n₂], which is real for ALL positive integers n₁ and n₂, with no restriction needed between them.
[1996]