# Prove that the language $L_1 = \{a^ib^{2i}c^j \;|\; i,j ≥ 0\}$ is context-free

Prove that the language $L_1 = \{a^ib^{2i}c^j \;|\; i,j ≥ 0\}$ is context-free.

I have a grammar like this but there are some strings that are not be able to be generated

\begin{align} S &\to aSbb \;|\; C \\ C &\to cC \;|\; \epsilon \end{align}

One example is the string $abbc$ cannot be generated.

You've almost got it. Consider that $L_1$ is simply the concatenation of $L_a$ and $L_c$, where $L_a = \{a^ib^{2i} \;|\; i ≥ 0\}$ and $L_c = \{c^j \;|\; j ≥ 0\}$, since the two parts of each sentence are independent of each other.
That should suggest a grammar which starts with the production $L\to AC$, and which then proceeds to define $A$ and $C$ in the obvious way.