Final answer:
To find the output signal y[n] for the given system, convolve the system's impulse response h[n] with the input signal x[n].
Step-by-step explanation:
To find the output signal y[n] for the given system, we need to convolve the system's impulse response h[n] with the input signal x[n]. The convolution operation is denoted by the asterisk (*) symbol. Let's calculate the convolution:
y[n] = (h * x)[n] = (∑[k=-∞ to ∞] h[k] x[n-k]) = (∑[k=-∞ to ∞] ((1/3)^k u[k]) (1/4)^(n-k)) = (∑[k=-∞ to ∞] (1/3)^k (1/4)^(n-k) u[k])
However, since the input signal x[n] is only defined for n ≥ 0, the sum can be simplified as:
y[n] = (∑[k=0 to ∞] (1/3)^k (1/4)^(n-k) u[k])