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For this question the guide says to use $Y(e^{j\omega}) = H(e^{j\omega}) X(e^{j\omega})$. I have been able to find the discrete time Fourier transform of the impulse function as $$H(e^{j\omega}) = \frac{1}{1-\frac12 e^{-j\omega}}$$ Finding the same for the impulse function is what I do not understand.
Note that the given definition of $x[n]$ is undefined for $n=0$. But even if you changed it by replacing the unit step $u[n]$ by the shifted step $u[n-1]$ to exclude the value $n=0$, then the problem would still be very hard to solve. So it's more than likely that the correct definition of $x[n]$ has no $n$ in the denominator, and the current form is just a consequence of a typo or of a misinterpretation.
