Need critique and proof of a theorem

• H.B.
In summary, the discussion is about a new theorem that the participants are trying to prove. The theorem involves a function that is defined recursively and has a limit as n approaches infinity. The participants have tested the theorem with different values of a and b and have not found a counterexample. The theorem was inspired by a physical formula about kinetic energy.

H.B.

Can anyone prove the next theorem, if it's new I think it's mine.
Thank you for trying.

Definition

a en b are integers a>0, b>0

$$X1$$ f(1) =a
$$X2$$ f(n+1) = (2f(n))mod(a+b)

Theorem
$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}|{f(k+1)-f(k)}|\left(\frac{1}{4}\right)^k = \frac{ab}{2(a+b)}$$

In Mathematica I can give different values for a and b, like a=8, b=9 and this is not a counterexample. n must be a large number like n=50 or n=100. If there is a single counterexample then this is enough to prove the theorem is wrong.

Nice observation, I don't know if this is already known. Anyways it seems to be true

How did you come up with that?

HB - I erased my post very shortly after making it as I realized in my program I had the formulas all wrong. I now have it correctly entered and It appears to be true for all a and b up to 10k (not that that means it will always be true )

Pere, I came up with the theorem when I defined a physical formula about kinetic energy.
$$E(t)=\frac{m_1*m_2}{2(m_1+m_2)}V(t)^2$$

1) What is a theorem?

A theorem is a statement that has been proven to be true using logical reasoning and existing mathematical principles. It is a fundamental concept in mathematics and science, and serves as the basis for many mathematical proofs and theories.

2) Why do we need to critique and prove a theorem?

Critiquing and proving a theorem is important because it allows us to ensure the validity and accuracy of the statement. By carefully examining the logic and assumptions used in the proof, we can verify that the theorem is indeed true and can be applied in various contexts.

3) How do you critique a theorem?

To critique a theorem, one must carefully analyze the proof and look for any weaknesses or flaws in the logic. This may involve checking the assumptions and premises used, looking for counterexamples, and verifying the steps of the proof. It is also important to consider alternative approaches and perspectives to ensure the theorem holds true in different contexts.

4) What is the role of peer review in critiquing and proving a theorem?

Peer review is a crucial step in critiquing and proving a theorem. It involves having other experts in the field carefully examine the proof and provide feedback and suggestions for improvement. This helps to identify any potential errors or weaknesses in the theorem and ensures that it is thoroughly vetted before being accepted as true.

5) Can a theorem be proven wrong?

Yes, a theorem can be proven wrong if there is a counterexample or if the logic and assumptions used in the proof are flawed. This is why it is important to thoroughly critique and review a theorem before accepting it as true. However, once a theorem has been rigorously proven and accepted by the scientific community, it is generally considered to be true and reliable.