Hi, Narbe,
the question defines by induction a sequence of integers in $\displaystyle \{0,\ldots,6\}$: $\displaystyle x_0=3$ and, for $\displaystyle n\geq 0$, $\displaystyle x_{n+1}=4x_n+1 ({\rm mod }7)$. So $\displaystyle x_1=4\cdot 3+1=13=6 ({\rm mod }7)$: $\displaystyle x_1=6$. And so on. The question is: give an expression for $\displaystyle x_n$ for all $\displaystyle n$.
My advice would be to compute a few terms of the sequence and look at what happens.
The "pseudo-random numbers" thing is a reference to a well-known method of getting random numbers (or kind of) on a computer, that works with similar inductive equations (but with larger numbers than 4 and 7). To learn more (not for your exercise but personal interest), look at
this.