Final answer:
The response of the first-order system h[n]=(−5/6)ⁿu(n) is (-5/12)ⁿcos(πn/4)u(n) for the input x[n]=(1/2)ⁿcos(πn/4), and (-5/12)ⁿcos(2πn/3)u(n) for the input y[n]=(1/2)ⁿcos(2πn/3)u[n].
Step-by-step explanation:
The response of the first-order system h[n]=(−5/6)ⁿu(n) to each input can be found by substituting the given inputs into the system equation.
For the first input, x[n]=(1/2)ⁿcos(πn/4), we have h[n]=(−5/6)ⁿu(n), so the response is h[n] = (−5/6)ⁿ(1/2)ⁿcos(πn/4)u(n) = (-5/12)ⁿcos(πn/4)u(n).
For the second input, y[n]=(1/2)ⁿcos(2πn/3)u[n], the response is h[n] = (−5/6)ⁿ(1/2)ⁿcos(2πn/3)u(n) = (-5/12)ⁿcos(2πn/3)u(n).