Tricky FE
Find all functions $f:\mathbb{N}\mapsto\mathbb{Q}$ satisfying \[f(n+1)=f(n)+\dfrac{1}{n^2+5n+6}\] for all natural numbers $n$.
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
-
- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Re: Tricky FE
Fix $f(1)$ any rational number. Then all other $f(n)$ are uniquely determined by the given relation and all of them must also be rational. Thus there are infinitely many solutions.
Re: Tricky FE
What are the solutions?
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
-
- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Re: Tricky FE
$f(n)=f(1)+ \frac {1}{2.3} +\frac {1}{3.4} + \frac {1}{4.5}+....+\frac{1}{(n+1)(n+2)}= f(1)+\frac{1}{2}-\frac{1}{3}+\frac {1}{3}-\frac{1}{4}+\frac{1}{4}+....+\frac{1}{n+1}-\frac{1}{n+2}$=$ f(1)+\frac{1}{2}-\frac {1}{n+2}$. And $f(1)$ is determined by our choice.
P.S: I don't see any 'tricky' part.
P.S: I don't see any 'tricky' part.
Re: Tricky FE
\[f(n+1)=f(n) + \frac 1{n+2} - \frac 1{n+3}\]
Telescope.
Telescope.
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Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
-
- Posts:107
- Joined:Sun Dec 12, 2010 10:46 am
Re: Tricky FE
This part is a text book problem.
Re: Tricky FE
The trick is the equation can be rewritten as \[f(n+1)+\dfrac{1}{(n+1)+2}=f(n)+\dfrac{1}{n+2}\] so \(f(n)+\dfrac{1}{n+2}\) is constant. Letting the constant \(c\in\mathbb{Q}\) and solving leads to \(f(n)=c-\dfrac{1}{n+2}\).
My intention was to show the idea of constantifying which works for a large class of FEs. See the next FE.
My intention was to show the idea of constantifying which works for a large class of FEs. See the next FE.
- What is the value of the contour integral around Western Europe?
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.
- Zero.
- Why?
- Because all the poles are in Eastern Europe.
Revive the IMO marathon.