SOLUTION: Prove that (z^n) + (z*)^n is always real for integer n

Question 1181500: Prove that (z^n) + (z*)^n is always real for integer n
Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
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Prove that (z^n) + (z*)^n is always real for integer n
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            In this problem, it is assumed (hidden assumption) that z is a complex number   z = a + bi.
            Another hidden assumption is that z* is the complex conjugate to z.

In the binomial expansion of  z^n = ,  the imaginary parts (addends) are those and only those 

that contain the factors  (ib)  in ODD  degrees.



Similar is for the binomial expansion of  (z*)^n =  :  the imaginary parts (addends) are those and only those 

that contain the factors  (-ib)  in ODD  degrees.



But the corresponding imaginary addends in binomial expansions  z^n  and  (z*)^n  go with opposite signs, and, THEREFORE,

they cancel each other in the sum  z^n + (z*)^n.



Therefore, the sum  z^n + (z*)^n  is a real number.

Solved, answered and explained.

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