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If z is a complete number satisfying    

              z+1/z=1,  

calculate    

              z^10+1/(z^10).

 Aug 2, 2016
 #1
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I assume that you mean complex number rather than complete number.

 

There are two obvious methods, first, you can build upto the result by a series of multiplications and squarings.

So for starters (z+1/z)^2 = z^2 + 2 + 1/z^2 = 1,  so  z^2 + 1/z^2 = -1.

Squaring that, z^4 + 2 + 1/z^4 = 1, so z^4 + 1/z^4 = -1 and so on.

The intermediate expression z^3 + 1/z^3 can be obtained by multiplying z + 1/z by z^2 + 1/z^2 etc..

 

A second method requires you to solve the equation  z + 1/z = 1.

That gets you z = 1/2 +- i*sqrt(3)/2 = cos(pi/3) +- i*sin(pi/3) and now you can use De'Moivre's theorem.

 Aug 3, 2016

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