= Solution
A <context-free grammar> is a quadruple $(V,\Sigma,P,S)$ consisting of finite nonterminal and terminal alphabets, a set of productions $A\to\alpha$ with one nonterminal $A$ on the left, and a start symbol $S$. A <context-free language> is a language generated by such a grammar.
The <pumping lemma for context-free languages> says that there is a pumping length $p$ such that every $w$ in the language with $|w|\geq p$ can be written
$$
w=uvxyz
$$
with $|vxy|\leq p$, $|vy|>0$, and $uv^ixy^iz$ in the language for every integer $i\geq0$.
The language in (i) is not context-free. If it were, pump
$$
w=a^p b^{2p}c^p.
$$
The substring $vxy$, having length at most $p$, cannot meet both the $a$- and $c$-blocks. If pumping changes the number of $a$'s or $c$'s, those two counts cease to agree. If it changes neither, it changes the number of $b$'s while the outer counts remain $p$, so the middle count ceases to be $2p$. Every permitted decomposition therefore fails, contradicting the pumping lemma.
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