Question

Find the value of i^n , when n is an odd integer.​

Answer 1

Answer:

Step-by-step explanation:

Let’s first assume that  n  is a positive odd integer.

(−1)n=(−1)⋅(−1)(n−1)=(−1)⋅(−1⋅−1⋅…⋅−1(n−1) times)  

According to our first assumption  (n−1)  must be even, therefore we can split up the products into pairs, like this:

(−1)n=(−1)⋅((−1⋅−1)⋅(−1⋅−1)⋅…⋅(−1⋅−1))=  

(−1)⋅(1⋅1⋅…⋅1)=(−1)⋅1=−1  

But we’re not quite done. What happens if  n  is a negative odd integer?

Here we need to use the fact that  x−n=1xn=(1x)n  for any  x≠0  

Setting  x=−1⟹(−1)−n=(1−1)n=(−1)n

Answer 2

Answer:

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