Is this conclusion via rules of inference correct?

Question

Use rules of inference to show:

1. ∀x(P(x) → Q(x)) premise
2. ∀x(Q(x) → R(x)) premise

3. ¬R(a) premise

4. ¬P(a) conclusion

I have a lot of trouble with these sort of questions and was wondering if I did this correctly. Usually I have no idea which rules to use and it feels like I just need to try them randomly until they work.

1) ∀x(P(x) → Q(x)) premise

2) P(a) → Q(a) Universal instantiation (1)

3) ∀x(Q(x) → R(x)) premise

4) Q(a) → R(a) Universal instantiation (3)

5) P(a) → R(a) Hypothetical syllogism (2,4)

6) ¬R(a) premise

7) ¬P(a) Modus tollens (5,6)

Answer 1

Yes this looks perfectly good to me.

Answer 2

I dont know the formal name of the different silogism but you can write

$$\forall x(P(x)\implies R(x))\tag{1}$$

To see this clearly you can prove it via "if $x\in P(x)$ then..." and use the notation:

$$(A\implies B)\iff(\lnot A\lor B)$$

Then if $\lnot R(a)$ and $(1)$ is true then $\lnot P(a)$ must be true.