Re: worm problem,PEDAGOGY

[ntganiel@weizmann.weizmann.ac.il (Uri Ganiel)]

> Hi Uri:
> I finally looked into your posted solution (of the Leigh's worm problem).
> I compared your non-interative expression for the distance covered by
> the bug in n steps
>                      x{n} = (n+1)*(sum from k=1 to k=n of (1/k))
>
> with the iterative formula I was able to derive (posted yesterday).
>
>                      x{n}=[x{n-1}+1]*(1+1/n)
>
> They lead to identical answers. How did you reason to obtain your
> formula from the wording of the problem? How did you recognize the
> harmonic series (1+1/2+1/3+1/4+ ...+1/n) in it?
>                                                  Ludwik Kowalski

Ludwik,
These things are easier seen on paper (I just looked and recognized it),
than on e-mail where one is limited with fonts and symbols. Let me try,
though: write it out in detail, for a few cases (n=1, 2, 3, 4 ,etc.) and
you easily deduce the pattern for a general n, to be:

x(n)=(...((((2/1+1)*3/2+1)*4/3+1)*5/4+1)*6/5+1......)*((n+1)/n)

Write it on paper - so much more convenient. Then open it up, going from
right to left. Do you see what happens? you have (n+1) multiplying
everything. Then you have 1/n multiplying two terms: 1 and everything else.
1/n times "everything else" is  1/n times n/(n-1) which gives you 1/(n-1),
and that now multiplies 1 plus "everything else" again. You see how awkward
I am becoming in my explanation, but if you just write it down you will see
it immediately. You continue to open the parentheses, and the result pops
out as I wrote it down.
I hope this helps.

Best                    -                       Uri

Prof. Uri Ganiel
Head, Department of Science Teaching
The Weizmann Institute of Science
Rehovot 76100, ISRAEL

Phone: 972-8-9343894
FAX:    972-8-9344115

E-mail:  NTGANIEL@weizmann.weizmann.ac.il