# Is the language $\{a^n b^m \mid 2n + 3m \le 1000 \}$ regular?

We have a language $$L = \{a^n b^m \mid 2n + 3m \le 1000 \}$$

Is this language regular?

I'm trying to disprove this using the Pumping Lemma, but it didn't work.

assume I say x = $$x=a^{h}$$ and $$y=a^{t}$$ and $$z =a^{n-t-h}b^m$$

if I say i = 0 everything is okay because $$L =a^{n-t}b^m$$ and 2(n-t) + 3m <= 1000

if I say i = 2 $$L =a^{n+t}b^m$$ and 2(n+t) + 3m <= 1000 because I'm not sure about t value.

I think it didn't work. Is this language regular? How can I prove that?

To see that the language is finite, notice that the maximum length of each word in $$L$$ is upper bounded by $$500$$. Indeed, if $$w \in L$$, and $$n$$ (resp. $$m$$) is the number of $$a$$s (resp. $$b$$s) in $$w$$:
$$2|w| = 2(n + m) \le 2n+ 3m \le 1000.$$
• @narges Hint: you can rewrite $2n-3m \le 1000$ as $n \le 500 + (3/2)m$. Can you come up with a word in the language that can be be "pumped" enough times to violate that inequality? – Steven Apr 13 at 11:26