# Proving the regularity of the following language

I have a question about the following problem:

Prove that the language $$\{a^nva^n | v \in \Sigma^*, n \ge 1\}$$ is regular over $$\Sigma = \{a,b\}.$$

I know that in proving a language is regular I can either construct a DFA, give a regular expression or show the Nerode-Relation has finite index.

My main problem is, that I cannot understand why this should be a regular language, since $$\{a^nb^n | n \ge 1\}$$ is not regular and of similar form. Also, I do not understand how a DFA should recognize the same amount of a's at the front and at the end of a word.

• It doesn't have to make sure there are the same number of $a$ at both ends. Sufficient that there is one. Nothing says $v$ can't start or end with an $a$. – rici Feb 12 at 21:56

You can always take $$n = 1$$ for any string in your language...