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Question
If `((1 + i)/(1 - i))^x` = 1, then ______.
Options
x = 2n + 1
x = 4n
x = 2n
x = 4n + 1, where n ∈ N
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Solution
If `((1 + i)/(1 - i))^x` = 1, then x = 4n.
Explanation:
Given that: `((1 + i)/(1 - i))^x` = 1
⇒ `(((1 + i)(1 + i))/((1 - i)(1 - i)))^x` = 1
⇒ `((1 + i^2 + 2i)/(1 - i^2))^x` = 1
⇒ `((1 - 1 + 2i)/(1 + 1))^x` = 1
⇒ `((2i)/2)^x` = 1
⇒ (i)x = (i)4n
⇒ x = 4n, n ∈ N
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i2 + i3 + ... + i4000 =
