Division in 3 parts

[Function]

Hi everyone

 

Imagine a horizontal line of length 1. Divide it into 2 equal parts. Now pick a random side of the 2 equal parts: left, or right. If you pick the left half, divide it in 2 and now focus on the 'new' right half (of length 0,25). Divide that into 2 and now focus on the 'new' left half. Divide that into 2 and focus on the 'new' right side.

 

As you see, alternating between 'newly cut' left and right sides and divide that once again in 2.

 

I find that using this algorythm approaches 2 values: 1/3 for the left side, and 2/3 for the right side.

 

(1) Is this true and (2) how can this be proven?

 

Thanks.

 

F.

Edited June 24, 2015 by Function

[imatfaal]

if you imagine the line as length 1

the first cut is at 1/2

the second at 1/4

3/8, 5/16, 11/32...

 

that is to say

 

[latex]\frac{\frac{2^n-(-1)^n}{3}}{2^n}[/latex]

 

separate the 1/3

 

[latex]\frac{1}{3} \cdot \frac{2^n-(-1)^n}{2^n}[/latex]

 

The second term clearly gets damn close to one as n grows - so the whole expression tends to 1/3

[Function]

if you imagine the line as length 1

the first cut is at 1/2

the second at 1/4

3/8, 5/16, 11/32...

 

that is to say

 

[latex]\frac{\frac{2^n-(-1)^n}{3}}{2^n}[/latex]

 

separate the 1/3

 

[latex]\frac{1}{3} \cdot \frac{2^n-(-1)^n}{2^n}[/latex]

 

The second term clearly gets damn close to one as n grows - so the whole expression tends to 1/3

 

Ah, thanks, imatfaal