The language is $L = \{a^{i} b^{j} c^{k} \;|\; k \neq 2j\}$. I'm trying to write a grammar for this language, what I have so far is:
$S \rightarrow AT_{1} \;|\; AT_{2} \;|\; AT_{3} \;|\; AB \;|\; AC$
$A \rightarrow aA \;|\; \varepsilon$
$B \rightarrow bB \;|\; \varepsilon$
$C \rightarrow cC \;|\; \varepsilon$
$T_{1} \rightarrow bbB'T_{1}c \;|\; \varepsilon $ (for $2j > k$)(1)
$B' \rightarrow bB' \;|\; b$
$T_{2} \rightarrow bT_{2}ccC'\;|\; \varepsilon$ (for $2j < k$)
$C' \rightarrow cC' \;|\; c$
$T_{3} \rightarrow bT_{3}c \;|\; \varepsilon$ (for $j = k$)
the problem that I am having is, the string $bbccc$ can't be generated although valid, in that case $j = 2$ and $k = 3$ so $2\times 2 > 3$ corresponding to production rule (1), how can I fix this?