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Question
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
Solution
\[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n = \left( 1 - i \right)^n \left( 1 - \frac{i^4}{i} \right)^n [ \because i^4 = 1]\]
\[ = \left( 1 - i \right)^n \left( 1 - i^3 \right)^n \]
\[ = \left( 1 - i \right)^n \left( 1 + i \right)^n [ \because i^3 = - i]\]
\[ = \left[ (1 - i)(1 + i) \right]^n \]
\[ = (1 - i^2 )^n \]
\[ = 2^n [ \because i^2 = - 1]\]
Thus, the value of
\[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\] is 2n.
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