Need critique and proof of a theorem

[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
<br /> \lim_{n\rightarrow\infty}\sum_{k=1}^{n}|{f(k+1)-f(k)}|\left(\frac{1}{4}\right)^k = \frac{ab}{2(a+b)}<br />

 

[H.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.

 

[Pere Callahan]

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

How did you come up with that?

 

[Diffy]

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 )

 

[H.B.]

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
I think this is not the right forum to start a discussion about this.