A rat runs in labyrinth of concentric circles of increasing diameters 1m,3m,5m,7m...?

Question

connected by radial passages.

The rats starts at North from the center,
runs 1m cw, turns left, continues to the next circle,
runs 1m ccw, turns right, continues to the next circle,
runs 1m cw, turns left, continues to the next circle,
runs 1m ccw, turns right, continues to the next circle,
etc..

In what direction can we see the rat after long time?

Answer 1

The formula for the circumference of a circle is C=PI*D.

The first circle has diameter 1, so

    circumference = PI

Since the rat travels 1 meter, that is

    1m/3.14m = .318

of the 360 degrees of the full circle.

    .318*360 = 114.6 degrees.

Turning back along the circle which is 3 meters
in diameter, he moves 1 meter of the total of
3*PI = 9.4 meters.

    1/9.4 * 360 = 38 degrees.

Since he started at 114 clockwise from "north"
and moves back 38 degrees, he's now at 76
degrees (CW from north).

Similarly for the 5 meter circle, he moves
CW by 22.9 degrees, leaving him at 99 degrees.

Going counter clockwise on the 7 meter circle,
he subtracts 16.37 degrees, leaving him at 82.9.

Emperically, the pattern is:

    diameter 1 angle= 114.592
    diameter 3 angle= 76.394
    diameter 5 angle= 99.313
    diameter 7 angle= 82.942
    diameter 9 angle= 95.675
    diameter 11 angle= 85.257
    diameter 13 angle= 94.072
    diameter 15 angle= 86.433
    diameter 17 angle= 93.173
    diameter 19 angle= 87.142
    diameter 21 angle= 92.599
    diameter 23 angle= 87.617
    diameter 25 angle= 92.200
    diameter 27 angle= 87.956
    diameter 29 angle= 91.908
    diameter 31 angle= 88.211


In more general terms the angle (from "north") is:

    angle = 360/PI * (1 - 1/3 + 1/5 - 1/7 + ...)

The series in parenthesis converges to PI/4, thus:

    angle = 360/PI * PI/4 = 360/4 = 90 degrees (CW from north)

Therefore, he will eventually be heading due "east".

Answer 2

2-2/3+2/5-2/7+2/9-2/11+2/13 -2/15 + 2/17....2/(2n-1).
Find the sum of this series and you will have the number of radians cw from North that the rat is located.

Answer 3

That way...