Re: Trisecting and angle after infinite steps


<tuanglen@xxxxxxxxxxx> wrote in message
news:1116465122.156862.302590@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> I'm trying to solve an infinite series problem, but I'm stuck. It's not
> a deep mathematical mystery. It's just a problem in a calculus book I'm
> reviewing. It has been more than 20 years since my last "homework"
> problem, but I try to review this stuff from time to time on my own so
> I don't lose it as most of my classmates from those days have long
> since done.
>
> [This is from p. 651 in Howard Anton's new 8th ed. Calculus, which is
> generally a very good book for self study]:
>
> "Suppose an angle theta [the one in the diagram has its 'initial edge'
> at 3 o'clock if you imagine a clock face and its other at roughly 11
> o'clock] is bisected to produce ray R1 [shown going from the origin off
> in the roughly 1 o'clock direction]. Then the angle between R1 and the
> initial side is bisected to produce ray R2 [shown as a ray going off in
> the roughly 2 o'clock direction, or theta/4.] Thereafter, rays R3, R4,
> R5... are constructed in succession by bisecting the angle between the
> preceding two rays [R3 is drawn between R1 and R2 at about 1:30, R4
> between R2 and R3, etc. converging on a limit ray at theta/3]. Show
> that the sequence of angles that these rays make with the initial side
> has a limit of theta/3."
>
> Okay, if I'm calculating correctly the angle of these rays, expressed
> as a fraction of theta, creates the following sequence:
>
> 1/2, 1/4, 3/8, 5/16, 11/32, 21/64, 43/128, 85/256, 171/512, ...
>
> This can be expressed as a recurrance relation where each term is the
> average of the previous two (using parentheses to denote subscripts):
> X(n) = [ X(n-2) + X(n-1) ]/2
> initialized by the first two terms 1/2 & 1/4.

Well, this is a standard problem. You start by substituting X(n) = a^n into
the above equation. That gives you (after rearranging):
2 a^2 - a - 1 = 0.
This quadratic equation has the two solutions a1 and a2, where a1=1 and
a2=-1/2.

Then the complete solution to your problem above is:
X(n) = c1 * a1^n + c2 * a2^n,
where c1 and c2 are constants independent of n. They are determined from the
first two terms.

You'll find c1 = 1/3 and c2 = -1/3. So the solution may be written as

X(n) = 1/3 * (1 - (-1/2)^n).

-Michael.


.

Relevant Pages

  • Trisecting and angle after infinite steps
    ... It's just a problem in a calculus book I'm ... "Suppose an angle theta [the one in the diagram has its 'initial edge' ... o'clock] is bisected to produce ray R1 [shown going from the origin off ... If I could create a closed form expression for term X, ...
    (sci.math)
  • **SPOILERS** TNA iMPACT! Results For 12-17-09
    ... The lumberjacks in the match were the other six men in their ... Morgan had Ray set for the Hellevator, ... Terry versus A.J. Styles, Tomko, and Kurt Angle. ...
    (rec.sport.pro-wrestling)
  • Re: how to find wavelength in matlab
    ... but I once ran into an imaginary reflection angle. ... angle had to be complex once the ray entered the medium in order to ... which was set off on an axis colinear with the array axis. ... and excited the Scholte wave, which was what I was interested ...
    (comp.soft-sys.matlab)
  • Re: how to find wavelength in matlab
    ... but I once ran into an imaginary reflection angle. ... angle had to be complex once the ray entered the medium in order to ... which was set off on an axis colinear with the array axis. ... and excited the Scholte wave, which was what I was interested ...
    (comp.soft-sys.matlab)
  • TNA Impact spoilers - Pole Match Warning!
    ... AJ comes out to interrupt and Angle tells him to come to the ring and say it to Sting's face. ... AJ says for the rest of the Mafia to step out of the ring and they do. ... *Backstage Ray is telling the Frontline he needs them all to leave so he can take care of Angle tonight and he believes Angle will do the same. ...
    (rec.sport.pro-wrestling)