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

Mathematics
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ok, first few terms come out as: 1, 3, 11, 41, 153
\[a_n=4a_{n-1}-a_{n-2}\]
\[a_n=\frac{\sqrt{5}+5}{10}(2+\sqrt{5})^n+\left(\frac{1}{2}-\frac{\sqrt{5}}{10}\right)(2-\sqrt{5})^n\]

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Other answers:

Looks like I'm a little off.
I think I see my error
ok...one little plus sign messed it up \[a_n=\frac{\sqrt{3}+3}{6}(2+\sqrt{3})^n+\left(\frac{1}{2}-\frac{\sqrt{3}}{6}\right)(2-\sqrt{3})^n\]
messed up my calculations that is :)
I arrived at my result by squaring the expression for \(a_{n+1}\) and also the expression for \(a_n\), then combining them both.
and I used your result to get mine :)
:)
Zarkon - I am convinced that your brain lives in another parale universe! :D
could be ;)
Wow! I just checked Zarkon's result and it is actually correct - not that I had any doubt of course ;-)
lol
there are techniques to solve these kinds of problems.
would you be able to give any helpful pointers to the types of topics to study for these problems?
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
how do you leap from the first equation to the second? the first one involves terms in n, n-1 and n-2?
Look at this ... http://en.wikipedia.org/wiki/Recurrence_relation#Solving
does this come under "Number Theory"?
oh - ok - thanks for the link - I love learning new things! :)
they are related to differential equations
really - that is very interesting. thanks again Zarkon for letting us "peek" a little inside your brain. :D
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.)
ok - I have plenty of reading material now. thanks again! and thanks to Mr.Math for posing such a question!
Thanks Zarkon! You're the best. I will have a look at the link you posted above. And thanks to asnasser as well.
@Zarkon: I'm looking for good textbooks on PDE's, could you recommend one or two to me?
I've never studied PDE's. I've worked with ODE's and Stochastic differential equation but not PDE's
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?
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.

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