Did I do this problem correctly?

Question:

Construct context-free grammars that generate the following languages (give your answers as a set of rules):

1. {a^kw|w{a,b},|w|=k,k0}

Answer:

S | A

A | aAB

B a | b

Possible strings to test on:

k = 1

-> |w| = 1 -> aa | ab

SAaAB aB aBaa | ab

k = 2

-> |w| = 2 -> aaaa | aabb | aaab | aaba

SAaAB aaABBaaBBaaBBaaaa aabb aaab | aaba