Recursive formula find closed something

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February 27th 2010, 05:17 PM
Henryt999

Recursive formula find closed something

Hi!
There is this problem on recursive formulas I do not understand.
The problem is:
examine the recursive: $a_{n+1}=\frac{a_n}{1+a_n}$ where $a_0 =\frac{1}{2}$ and $n=0,1,2,3,4...$
Find a closed formula of the type $a_n = f(n)$ where $f(n)$ is a expression that depends on n but not on for example $a_{n-1}$

I can probably do it if I would understand what they want, closed formula????
No idéa...where to start..
February 27th 2010, 05:22 PM
NonCommAlg

Quote:

Originally Posted by Henryt999

Hi!
There is this problem on recursive formulas I do not understand.
The problem is:
examine the recursive: $a_{n+1}=\frac{a_n}{1+a_n}$ where $a_0 =\frac{1}{2}$ and $n=0,1,2,3,4...$
Find a closed formula of the type $a_n = f(n)$ where $f(n)$ is a expression that depends on n but not on for example $a_{n-1}$

I can probably do it if I would understand what they want, closed formula????
No idéa...where to start..

they want you to prove that $a_n=\frac{1}{n+2}.$
February 28th 2010, 01:32 AM
HallsofIvy

Quote:

Originally Posted by Henryt999

Hi!
There is this problem on recursive formulas I do not understand.
The problem is:
examine the recursive: $a_{n+1}=\frac{a_n}{1+a_n}$ where $a_0 =\frac{1}{2}$ and $n=0,1,2,3,4...$
Find a closed formula of the type $a_n = f(n)$ where $f(n)$ is a expression that depends on n but not on for example $a_{n-1}$

I can probably do it if I would understand what they want, closed formula????
No idéa...where to start..

This problem tells you what a "closed formula" is! It "is an expression that depends on n" only. In other words, what you have always thought of, before, as a formula.

If $a_0= \frac{1}{2}$ , then $a_1= \frac{\frac{1}{2}}{1+ \frac{1}{2}}$ $= \frac{\frac{1}{2}}{\frac{3}{2}}= \frac{1}{2}\frac{2}{3}= \frac{1}{3}$ .

Then $a_2= \frac{\frac{1}{3}}{1+ \frac{1}{3}}= \frac{\frac{1}{3}}{\frac{4}{3}}$ $= \frac{1}{3}\frac{4}{3}= \frac{1}{4}$ .

Now do you see where NonCommAlg got his formula?

Of course you will need to prove that is correct. I recommend proving it by induction. Recursive formulas are ideally suited for induction.