Find step response for LTI system represented by the following impulse response?
The given impulse response is unbounded for $n\rightarrow -\infty$, and, consequently, the corresponding step response is also unbounded for all $n$, as you've shown. The step response given in the answer corresponds to a totally different impulse response:

$$h[n]=2^nu[n]\tag{1}$$

which is also unbounded for $n\rightarrow\infty$, but since the corresponding system is causal, the step response assumes finite values for finite $n$.