"Guess the next term" is very often not a proper mathematics problem, and I would agree with your teacher that it is not a proper mathematics problem in this case (but a diverting puzzle among friends).
However, the sequence has surprising properties about the limit proportions of 1s and 2s that would be "true" mathematics problems for any mathematician and I invite you to read up on this on the internet.
The instinctive and understandable disdain to any "guess the next term"-question is often clouding the judgment of people on the general interest of such a sequence.
Yet, calling the opinion of the teacher "belittling", your brother "bored" and asking for a "riposte" does not make it entirely believable that your brother asked this question out of enthusiasm. It is totally appropriate to call the guess the next term- question not a real maths problem.
I have some empathy for this situation, because my brother had a teacher who would not believe that the harmonic series diverges and we could not convince him otherwise, but your teacher did not say anything mathematically wrong at all, it is understandable that he did not like this puzzle and your entitled demands for "praise" are really not appropriate.
You and your brother are very privileged to be students in the times of the internet, why do you need your teacher to learn nice mathematics?
If your brother asked his teacher this puzzle to show him up then he should really, really let it go now. If he asked your teacher this puzzle to share his enthusiasm he might still try with the nicer properties of the sequence, but much better to find someone else to share his enthusiasm if an honest, but negative opinion seems belittling to your brother.