Is this simple PRNG secure? $G$ is a PRNG used in a stream cipher and defined in the following way:


G receives $s_0$ as an input, which is a random string drawn from a uniform distribution.
The output of step $i$ is $s_i = (s_{i-1}\cdot (N+1) + 1) \bmod{N}$, for $i=1,2,3\dots$


Is this PRNG secure? If not, what is the distinguisher?
$s_i = s_{i-1}\cdot(N + 1) + 1 = s_{i-1} \cdot N + s_{i-1} + 1$

but $s_{i-1} \cdot N = 0 \pmod N$, so

$s_i = s_{i-1} + 1 \pmod N$

which means you can discover the next number to be generated just looking to the current one...