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Mr.Math

Find \(a_n\) if \(a_0=1\) and \(a_{n+1}=2a_n+\sqrt{3a_{n}^2-2}\), \(n\ge 0\).

  • 2 years ago
  • 2 years ago

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  1. asnaseer
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    ok, first few terms come out as: 1, 3, 11, 41, 153

    • 2 years ago
  2. asnaseer
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    \[a_n=4a_{n-1}-a_{n-2}\]

    • 2 years ago
  3. Zarkon
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    \[a_n=\frac{\sqrt{5}+5}{10}(2+\sqrt{5})^n+\left(\frac{1}{2}-\frac{\sqrt{5}}{10}\right)(2-\sqrt{5})^n\]

    • 2 years ago
  4. Zarkon
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    Looks like I'm a little off.

    • 2 years ago
  5. Zarkon
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    I think I see my error

    • 2 years ago
  6. Zarkon
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    ok...one little plus sign messed it up \[a_n=\frac{\sqrt{3}+3}{6}(2+\sqrt{3})^n+\left(\frac{1}{2}-\frac{\sqrt{3}}{6}\right)(2-\sqrt{3})^n\]

    • 2 years ago
  7. Zarkon
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    messed up my calculations that is :)

    • 2 years ago
  8. asnaseer
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    I arrived at my result by squaring the expression for \(a_{n+1}\) and also the expression for \(a_n\), then combining them both.

    • 2 years ago
  9. Zarkon
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    and I used your result to get mine :)

    • 2 years ago
  10. asnaseer
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    :)

    • 2 years ago
  11. asnaseer
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    Zarkon - I am convinced that your brain lives in another parale universe! :D

    • 2 years ago
  12. Zarkon
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    could be ;)

    • 2 years ago
  13. asnaseer
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    Wow! I just checked Zarkon's result and it is actually correct - not that I had any doubt of course ;-)

    • 2 years ago
  14. Zarkon
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    lol

    • 2 years ago
  15. Zarkon
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    there are techniques to solve these kinds of problems.

    • 2 years ago
  16. asnaseer
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    would you be able to give any helpful pointers to the types of topics to study for these problems?

    • 2 years ago
  17. Zarkon
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    take \[a_n=4a_{n-1}-a_{n-2}\] and write it as \[x^2=4x-1\] \[x^2-4x+1\] find the roots of this simple quadratic. you will then have part of my answer above

    • 2 years ago
  18. asnaseer
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    how do you leap from the first equation to the second? the first one involves terms in n, n-1 and n-2?

    • 2 years ago
  19. Zarkon
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    Look at this ... http://en.wikipedia.org/wiki/Recurrence_relation#Solving

    • 2 years ago
  20. asnaseer
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    does this come under "Number Theory"?

    • 2 years ago
  21. asnaseer
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    oh - ok - thanks for the link - I love learning new things! :)

    • 2 years ago
  22. Zarkon
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    they are related to differential equations

    • 2 years ago
  23. asnaseer
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    really - that is very interesting. thanks again Zarkon for letting us "peek" a little inside your brain. :D

    • 2 years ago
  24. Zarkon
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    I guess I should say they are related to linear algebra ( both difference eq and differential equations can be solved, some of them at least, using linear algebra techneques.)

    • 2 years ago
  25. asnaseer
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    ok - I have plenty of reading material now. thanks again! and thanks to Mr.Math for posing such a question!

    • 2 years ago
  26. Mr.Math
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    Thanks Zarkon! You're the best. I will have a look at the link you posted above. And thanks to asnasser as well.

    • 2 years ago
  27. Mr.Math
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    @Zarkon: I'm looking for good textbooks on PDE's, could you recommend one or two to me?

    • 2 years ago
  28. Zarkon
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    I've never studied PDE's. I've worked with ODE's and Stochastic differential equation but not PDE's

    • 2 years ago
  29. Mr.Math
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    Oh, I didn't expect that. That brings to my mind another question, if I'm not bothering you. I'm a Math major, and I can't yet figure out what fields of Mathematics are more interesting to me. What can I do to find some areas of interest, which would also help me to choose my elective courses?

    • 2 years ago
  30. Zarkon
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    Just take as many classes as you can. I was going to do applied mathematics as a graduate student until I took a year long sequence in probability/statistics my senior year. You really don't know if you are going to like something until you fully immerse yourself into it.

    • 2 years ago
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