Context-free grammar for $a^{2n} b^{2n}$

Question

I have just started learning formal languages and here is a question I am facing a little hurdle:

Construct a context-free grammar for $\{ a^{2n}b^{2n} \mid n \ge 0 \}$.

This was what I got at first. $$S \to ab\mid aS\mid Sb\mid ab$$ Now I am getting this, $$S\to \epsilon$$

$$S\to aaSbb$$

$$G=(V,\Sigma,R,S)=(\{S,a,b\},\{a,b\},R,S)$$

$$R= \{S \to aaSbb\mid \epsilon\}$$ Is the approach to this question, right or is it done in a different way?

Answer 1

Here is a derivation in your grammar:

$$ S \to aS \to aab. $$

The word $aab$ does not belong to your language, hence your grammar is incorrect.

Answer 2

I think this is the solution: $$S\to \epsilon \mid aaSbb $$

Answer 3

the second grammer you wrote, is right. it can involve all strings belongs to the language with no extra string.