How can I obtain the response signal for this question?
HINT:

The (zero-state) step response is just the cumulative sum of the impulse response $h[n]$:

$$y_{ZS}[n]=u[n]\sum_{k=0}^nh[k]$$

This follows in a straightforward manner from the convolution of a unit step $u[n]$ with the impulse response:

$$y_{ZS}[n]=(h\star u)[n]$$