clever dagger
Hello everyone,
I’m currently working on problems involving mathematical induction. I understand the basic concept and can handle the most common types of problems since there are lots of tutorials focusing on them.
However, when it comes to more specific or unusual problems, I’ve had trouble finding similar examples with solutions or even any mention of them in tutorials.
If anyone could help me with some of the following problems, I’d really appreciate it — it would solve a lot of future issues (and probably prevent some exam consequences 😅).
Also, if you have any tips for learning university-level math more effectively, I’d love to hear them too.
Of course, I first tried to solve these using what we covered at school and looked through our materials, but the presentations aren’t great — the problems often don’t have solutions or even step-by-step examples.
During exercises, we usually only go through a small fraction of the types of problems that appear on exams.
Then I tried looking online for similar ones but didn’t find much. I even tried using AI, but I ended up correcting it most of the time — and sometimes I wasn’t sure if the solution was even correct.
Problems:
Prove that for n > 1:
(ii) 1/(n+1) + 1/(n+2) + ... + 1/(2n) > 13/24
(vii) 1/2^3 + 1/3^3 + ... + 1/n^3 < 1/4
Prove:
(i) 2^n > n^2, for n >= 5, n ∈ N
(v) 2^(n+2) > 2n + 5, for n >= 1, n ∈ N
Prove that for all positive real numbers a, b and for n > 1:
2^(n-1) * (a^n + b^n) > (a + b)^n
thick slate
lost badger
waxen mist