# How to formally prove that $f(n)=\Theta f(n+1)$

How to formally prove that $f(n)=\Theta f(n+1)$?

It's supposed to be easy, but I still can't get it. Thank you very much.

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Beware the abuse of notation there. –  Raphael May 19 '12 at 18:41

## 1 Answer

This depends on the sequence $(f(n))$. For example:

• If $f(n)=n!$ then $f(n)\ll f(n+1)$ hence $f(n)\notin\Theta(f(n+1))$.
• If $f(n)=\frac1{n!}$ then $f(n)\gg f(n+1)$ hence $f(n)\notin\Theta(f(n+1))$.
• If $f(n)=a^n$ with $a\ne0$ then $f(n)=\frac1af(n+1)$ hence $f(n)\in\Theta(f(n+1))$.
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oh OK then. I was wrong I thought they are Θ of each other. Thank you very much!!! –  Talom May 19 '12 at 15:45
This (abuse of) notation suggests that numbers can be in Landau relations. I think you should properly define the two sequences behind the scenes as to not confuse anybody. –  Raphael May 19 '12 at 18:41