NAU POTW-2011.09.20

I have found another POTW website, which is called northern Arizona university.

http://www.cefns.nau.edu/Academic/Math/departmentActivities/POTW/

The problem says that:


A very tiny bug starts at the origin in the (x, y) plane. He walks to the right one unit and ends up at (1, 0). Then he makes a left turn and moves forward half a unit, reaching (1, 1/2). Then he turns left again and goes half again as far, reaching (3/4, 1/2). If he keeps doing this (turning left and going half again as far) again and again, what is his limiting position?


It maybe not easy to consider two dimension sequence, and also have the direction changing all the time. But the bug always turn left. That exactly the complex multiple ( multiple a number by i means turn the vector anti-clock wise in 90 degree.).

So the iterative function is z(n)=z(n-1)+(i/2)^(n-1). z(0)=0. 
z(n)-z(n-1) =(i/2)^(n-1) ....(n)
z(n-1)-z(n-2)=(i/2)^(n-2)....(n-1)
...
z(1)-z(0)=(i/2)^0 ...(1)


Add all the n equations get z(n) - z(0) = [(i/2)^n-1]/(i/2-1) . So z(n) = [(i/2)^n-1]/(i/2-1)
z(n) -> 1/(1-i/2), when n-> Infinity.
In coordinates expression is {4/5, 2/5}.


I plot the graph with Mathematica

lis = {Re[#], Im[#]} & /@ Table[((I/2)^n - 1)/(I/2 - 1), {n, 0, 18}];
ListPlot[lis, Joined -> True, PlotRange -> {{-0.1, 1.1}, {-0.1, 0.6}}]