Question 1181500
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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.



<pre>
In the binomial expansion of  z^n = {{{(a+bi)^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 = {{{(a-bi)^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.
</pre>


Solved, answered and explained.