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Basic IZT problem- simplifying problem before long division

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A Basic IZT problem

Guys, This is a basic IZT problem, i have been trying to solve it through long division method.
The problem is, if i simplify the problem ,then i get (4Z-2)/(-1+4Z^2) and then i go for the long division method, i don't get the answer.

whereas , if i directly go for the long division i get X(Z)= {1,-1/2,1/4,-1/8,....)

is there anything wrong is simplifying this problem before long division?

comments requested.
 

Re: A Basic IZT problem

\[X(Z)=\frac{1-\frac{1}{2}z^{-1}}{1-\frac{1}{4}z^{-2}}=\frac{1-\frac{1}{2z}}{1-\frac{1}{4z^{2}}}=\frac{\frac{2z-1}{2z}}{\frac{4z^{2}-1}{4z^{2}}}=\frac{2z-1}{4z^{2}-1}*\frac{4z^{2}}{2z}=\frac{2z(2z-1)}{4z^{2}-1}=\frac{2z(2z-1)}{(2z-1)(2z+1)}=\frac{2z}{2z+1}\]
 
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