Step-by-step explanation:
I think you made a typo in the first 15 letters.
given the rest of the information I am absolutely sure there is no "D" in the pattern ...
I believe the first 15 letters are
A B A C A A B A C A A B A C A
now we can see the structure of the pattern more clearly :
A B A C A
is repeated, over and over again.
A is appearing 3 times in one such segment.
as A appears in the total pattern 137 times (is that correct, or also a typo ?), it means we have
137/3 = 45.666666...
repeats of the 5-letter segment.
so, 45 full repeats and then 2 remaining As of the 46th repeat.
so, for the total number of letters we get 5 letters for each time of the basic segment :
45 × 5 = 225
and then we know we get 2 more As.
look at the basic segment again
A B A C A
what is the longest sequence of letters containing only 2 As ?
A B A C
that means max. 4 more letters.
so, the greatest possible number of letters in the total pattern is
225 + 4 = 229
