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Still one of my favorite problems. Feel free to attempt to solve it.

  1. Start at a 0,0

  2. Travel along the x axis in the positive direction a distance of 1

  3. Turn 90 degrees counterclockwise

  4. go forward 1/2 of the distance of the previous step

  5. Repeat steps 3 and 4 forever.

  6. how far from the origin to you end up?

Bonus points for finding an equation for

1.Start at a 0,0

2.travel along the x axis in the positive direction a distance of A

  1. Turn θ degrees counterclockwise

  2. go forward B times the previous distance walked

  3. Repeat steps 3 and 4 forever.

  4. how far from the origin to you end up?

Aspwil
  • 105

1 Answers1

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Use complex numbers.

The final position is, supposing $1>B\ge 0$,

$$A+ABe^{i\theta}+AB^2e^{2i\theta}+\cdots$$ $$=A(1+Be^{i\theta}+B^2e^{2i\theta}+\cdots)$$ $$=A\frac{1}{1-Be^{i\theta}}$$

The distance is $$\left|\frac{A}{1-Be^{i\theta}}\right|=\frac{A}{\sqrt{1-2B\cos\theta+B^2}}$$

For $A=1$, $B=1/2$, $\theta=\pi/2$, the distance is $$\left|\frac{1}{1-\frac{i}{2}}\right|=\frac{2}{\sqrt{5}}$$

velut luna
  • 9,961
  • Man, i did this back in ninth grade. My answer was was more complicated, I did not eve know what a complex number was back then – Aspwil Jun 18 '17 at 18:59
  • I am interested to know your method without using complex numbers. – velut luna Jun 18 '17 at 19:05
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    I actually made an entire YouTube video series about it, because it's fairly complicated, The Mystic spiral: https://www.youtube.com/playlist?list=PL6GmULYxljW8Wv8s2A8Xbe8ztj-emh4gh – Aspwil Jun 18 '17 at 19:08
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    To be complete: $|\frac1{1-i/2}|=\frac2{\sqrt5}=\frac{2\sqrt5}5$. – Akiva Weinberger Jun 18 '17 at 19:13
  • @AkivaWeinberger Thanks! – velut luna Jun 18 '17 at 19:13
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    And $\left\lvert\dfrac A{1-Be^{i\theta}}\right\rvert=\dfrac A{\sqrt{B^2-2B\cos\theta+1}}$, I believe. (You get that by multiplying $\dfrac A{1-Be^{i\theta}}$ by its conjugate, $\dfrac A{1-Be^{-i\theta}}$, and taking the square root.) – Akiva Weinberger Jun 18 '17 at 19:15