40+ years of math, a fair amount fairly advanced, the term "counterexample" doesn't "strike a bell".

Google tells me that counterexample is 3x more prevalent than disproof so you seem to be "right on the money".

The majority of my math has been application driven but the joy of just pursuing the rigor or intellectual discipline that abstract math contains still exists.

Glad to see you could contribute.

PS - I cannot see the bolded lines that you reference.

Matt is correct that let k = n is an incorrect substitution because the equation can't be shown to balance. n is already used on the left and I arbitrarily inserted it on the right.

And yes, a single counterexample does disprove a general theorem. However, relying on always being able to quickly find a specific integer that disproves a general theorem is a risky way of doing proofs.

It's not the approach I'd recommend counting on - or studying for - going into a final. What method did he use to find 3? Trial and error is great when it works, but is a horrible feeling in an exam when it doesn't.

What if the next question is "is there any solution set where the equation is true"? Would the answer be "not 3" or would there be more work involved?

This approach shows a problem with relying a specific counterexample being quickly identifiable (or unidentifiable). Using the approach of "the first step is to decide if the theorem can possibly be correct. Of course it cannot, because..."

What if you start with n=2? Can it possibly be correct? Yes. 8? Yes. 32? Yes. 488? Yes. 3? No.

What if there are no "disprove" questions on the final?

Thanks for telling me where it is.

Regarding the OP question of how to prepare for the exam. I'll suggest that developing the intellect is similar to developing muscle and that well structured mental exercise will literally both grow new brain cells and new inter-connections between cells.

So his (or her) best strategy is to develop healthy habits and to exercise their mind regularly.

Please post all the math remarks that you can. It's enjoyable to encounter the diversity of topics in OD.

Take care.

Thanks for your help. All I can say is this class has been the most confusing and difficult to date

Go study Physical Chemistry for a year; spend 4 months with the Schrodinger equation. I still hurt 35 years later.

Best to you with the exam.