What conjecture can you make about the 19th term in the pattern: a, b, b, a, b, b, a,…?

You can predict that the sequence that repeats is a,b,b

Then you can use the division by 3 to determine if a term will be a, b or c. 

If the residue of the division is 1, the term will be a; if the residue is 2 the term will be b, and if the residue is 0 the letter will be b.

For example, the residue of 4/3 is 1, and the 4th term is a, while the residue of 6/3 is zero and the 6th term is b.

Now find the 19th term. If you divide 19 by 3 you get residue 1.

Then the 19th term will be letter a.