Mandelbrot Set?!?

Algebra Level 4

Warning: The real difficulty of this problem should be level 5. Don't attempt to solve the explicit formula for $\{a_{n}\}$ .

This problem will give you a little taste of the complexity of the mandelbrot set , even if only defined on real numbers:

$\{a_n\}$ is a sequence such that $a_{1}=\alpha$ , $a_{n+1}=a_{n}^2+\beta$ , $\alpha,\ \beta \in \mathbb R$ .

Which value of $\beta$ will always satisfy $a_{10}>10$ regardless of what $\alpha$ is?

Try not to use some kind of computer program.

-2

\dfrac{1}{2}

\dfrac{1}{4}

-4

1 solution

Alex Burgess
Aug 20, 2019

Observing fixed points: $a_n = a_{n+1} = a_n^2 + \beta \implies (a_n - \frac12)^2 + \beta - \frac14 \implies a_n = \frac12 + \sqrt{\frac14 - \beta}$ , which has a solution $\leq 10$ for $-90 \leq \beta \leq \frac14$ .

Hence, for all values of $\beta$ in this range, there exist a fixed point $\leq 10$ .

$\frac12$ is the only solution outside of this range:

With $\beta = \frac12$ :

$a_1 \geq \frac12$

$a_2 \geq \frac34$

$a_3 \geq \frac{17}{16} > 1$

$a_4 > \frac32$

$a_5 > \frac94$

$a_6 > \frac{81}{16} > 5$

$a_{10} > 5^{16} > 10$

If I've done my quick math correctly, the end of the first line should be -90 rather than -90.25 (it makes no difference to the overall point). Very nice solution process!

David Richner - 1 year, 9 months ago

Whoops, thanks for flagging.:

$\frac12 + \sqrt{\frac14 - \beta} = 10 \implies |\frac14-\beta| = 9.5^2 = 90.25$ .

Alex Burgess - 1 year, 9 months ago

1/2 + (1/4 - (-90))^.5 = 1/2 + (1/4 + 90)^.5 = 1/2 + (90.25)^.5 = 1/2 + 9.5 = 10

David Richner - 1 year, 9 months ago

@David Richner – Oh yeah....

Alex Burgess - 1 year, 9 months ago

I'm actually quite impressed! I don't think there is a simple approach here, but you've still made it. In fact, you can eliminate another three choices by observing the fixed point first. It's way better than the official answer which involves hairy calculations.

Alice Smith - 1 year, 9 months ago

Thank you!

Alex Burgess - 1 year, 9 months ago

How do you know that there is a fixed point?

Ash Robson - 1 year, 9 months ago

In fact, this solution is not the essence of it.

Stephen Curry - 1 year, 7 months ago

Mandelbrot Set?!?

1 solution

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