worm problem/PEDAGOGY

[kowalskil@alpha.montclair.edu]

This morning Jack wrote:
>              x_(n+1) =  x_n + 1  +  x_n(1/(n+1))
My iteration formula is:
               x(n) = [x(n-1)+1] + [x(n-1)+1]*(n+1)/n
       or      x(n) = [x(n-1)+1]*(1+1/n)

Knowing that for n=1 ---> x(1)=2.0000 we have:
                 n=2  --> x(2)=(2+1)*(1+1/2)=4.5000
                 n=3  --> x(3)=(4.5+1)*(1+1/3)=7.3333, etc. 

Bug's locations, after n seconds, with respect to the road, are shown below
Average velocities and average accelerations were calculated from x(n). 

   n      x(n)     v(n)     a(n)
---------------------------------
   2     4.5000   2.5000   0.5000
   3     7.3333   2.8333   0.3333
   4     10.417   3.0833   0.2500
   5     10.417   3.0833   0.2000
   6     17.150   3.4500   0.1667
   7     20.743   3.5928   0.1428
   8     24.461   3.7179   0.1250
   9     28.290   3.8290   0.1111
  10     32.219   3.9290   0.1000
  ...    ......   ......   ......
  99    517.738   6.1774   0.0101
 199    1174.61   6.8330   0.0050
 999    7484.47   8.4845   0.0010
2000    16364.9   9.1784   0.0005

The age of the universe is too short (to solve this non-constant 
acceleration problem) by looping with the iteration formula, even 
on parallel supercomputers. But a mathematician is able to solve 
it on paper by using only a pencil. 
                                                   Ludwik Kowalski