Determine if a language is regular or not. Don't know how to start to prove.

Question

I'm trying to understand how to prove the following language:

$$L= \{a^nx \mid n \geqslant 0 \text{ and } x \in \{a,b,c\}^* \text{ and } |x| = n + 1\}$$

In my opinion it's regular but I can't find a way to prove. I'm aware about the pumping lemma but I can't find a start point.

Answer 1

Suppose that $L$ is regular. Since regular languages are closed under intersection, then so is $$ L \cap a^*b^* = \{a^nb^{n+1}\mid n \geqslant 0\} $$ Now, since you know about the pumping lemma, you should be able to conclude that this language is not regular.