$\{a^nb^2a^n \mid n \ge0\}$
I'm studying for my final and I came across this language. I haven't dealt with characters of the same length on opposite ends with something in between.
I came up with the following grammar, but I think it accepts strings that it should not such as: $abb, bba$, etc.
$$\begin{align} S&\rightarrow X\\ X&\rightarrow XbbX \mid \epsilon\\ A&\rightarrow aA \mid \epsilon \end{align}$$
In dealing with CFGs, it's often useful to work from the inside out. Clearly $$ S\rightarrow aSa $$ will generate all strings of the form $a^nSa^n$, for $n\ge 0$. Then all you have to do is allow the possibility that $S$ will also eventually generate $bb$, which we can do by the production $S\rightarrow bb$, giving the grammar $$ S\rightarrow aSa\mid bb $$ or, if you wanted to be explicit about the $bb$ central part, you could write $$\begin{align} S&\rightarrow aSa \mid T\\ T&\rightarrow bb \end{align}$$ The problem with your grammar is that it can't guarantee that there will be the same number of $a$s on each end of a generated string.
