Re: Fun With Tangens


Hauke Reddmann wrote:
> Given are four points 1,2,3,4.
> Let the t12 be the ascent of the line through 1 and 2.
>
> a) t12=1 t13=2 t14=3 t23=4 t24=5 t34=?
> b) t12=1 t13=2 t14=4 t23=8 t24=16 t34=?
> c) t12=1 t13=1/2 t14=1/3 t23=1/4 t24=1/5 t34=?
>
> --
> Hauke Reddmann <:-EX8 fc3a501@xxxxxxxxxxxxxx
> His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn

The answers are all the "next" in sequence.
This is remarkable and may relate to a geometric theorem but I cant
think of one.

If the series is f0 f1 f2 f3 f4 f5 and del52=f5-f2 etc then the
relationship for this to happen is
del51*del42*del30 = del40*del31*del52

for arithmetic progression del51=del40 etc so this is true.
for geometric progression (a*k^n) del51=del40*k etc and the powers of k
balance each side
for reciprocals (fx=1/gx) delxy is (gx-gy)/gxgy =
-delxy(g)/gxgy then the gxgy's balance out leaving the above equation
on the delg's (glibly).

equally obvious: you can add and/or multiply a suitable f by a
constant. eg 3+4*1, 3+4*2, 3+4*4 etc also 1+5/(3+4*1), 1+5/(3+4*2) etc
thus generating a host of rational solutions.

Do these combinations exhaust the possibilities?
JJ

.