The language L = {a^nb^jc^k : k = jn } can be shown to not be context-free by using the pumping lemma for context-free languages.
The pumping lemma states that if a language L is context-free, then there exists a number p (the pumping length) such that any string s in L of length greater than or equal to p can be divided into three substrings x, y, and z such that:
- |xy| <= p
- |y| > 0
- for all i >= 0, xy^i z is in L
If we take any string from L that follows the pattern a^nb^jc^k, the pumping lemma tells us that we can pump the middle substring y any number of times to get a new string still in L.
For example, let's take the string "a^2b^2c^2"
If we divide this string into x = "a", y = "a" and z = "b^2c^2", we can pump y any number of times to get a^2b^2c^2, a^3b^2c^2, a^4b^2c^2, ...
However, we can see that in all of these new strings the value of k is not equal to j*n. Which means that this string can't be in the language L.
Therefore, we can conclude that the language L = {a^nb^jc^k : k = jn } is not context-free.