Consider this question taken from Oppenheim - Discrete-time Signal and Systems:
Now consider its solution (from the Solutions manual)
My question is: is the solution for itens (a) and (b) valid? It considers a sequence with only one sample in both cases, how this can be a general solution? If it's valid, based on what?
The solution in (a) is correct because it was specified in the question itself that $x[x]=\delta[n]$, where $\delta[n]$ is the unit discrete impulse (i.e., Kronecker's Delta). The solution in (b), however, is not general per se and you are correct in this point.
The linearity of the system must be proved by giving the input $x[n] = a x_1[n] + b x_2[n]$ and verifying that the output is $y[n] = a y_1[n] + b y_2[n]$, where $y_1[n]$ is the output of $x_1[n]$ and $y_2[n]$ is the output of $x_2[n]$. Doing this requires some computation (since you need to do the recursion), but it can be done. However, there is a more clever way to do this: if you re-read the recursion as $y[n] = x[n] + ny[n-1]$, after little computation, we can observe that $$ y[0] = x[0] \\ y[1] = x[1] + x[0] \\ y[2] = x[2] + 2(x[1] + x[0]) \\ y[3] = x[3] + 3x[2] + 6(x[1]+x[0]) \\ y[4] = x[4] + 4x[3] +12x[2] +24(x[1]+x[0]) \\ y[5] = x[5] + 5x[4] +20x[3] +60x[2] + 120(x[1] +x[0]) \\ \cdots \\ y[n] = x[n] + n x[n-1] + n(n-1) x[n-2] + n(n-1)(n-2) x[n-3] + \dots + n!(x[1]+x[0]) $$ that is, $$ y[n] = \sum_{m=0}^{n} c_{m,n} x[n-m] $$ for appropriate constants $c_{m,n}$. And the linearity of the system is now clear and follows immediately.