Frequency Response Help
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The input signal in part (b) has 2 terms, and is the sum of 2 exponentials, one with frequency $$\omega_1 = 0$$ and the other with frequency $$\omega_2 = \pi/2$$

Since the system is linear, then the output is the sum of the output to each individual input. That is: $$y[n] = y_1[n] + y_2[n]$$

Also, since complex exponentials are eigenvectors of LTI systems, then the output will be the same exponential multiplied by the frequency response (i.e. the eigenvalue).

So,

$$
y_1[n] = 1 \times H(e^{j 0})
$$

and

$$
y_2[n] = e^{j n \pi/2} \times H(e^{j \pi/2})
$$

Where $$H(e^{j0}) = 1$$ and $$H(e^{j\pi/2})= 2+j$$.