- Mr.Math

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

- katieb

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- asnaseer

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

- asnaseer

\[a_n=4a_{n-1}-a_{n-2}\]

- Zarkon

\[a_n=\frac{\sqrt{5}+5}{10}(2+\sqrt{5})^n+\left(\frac{1}{2}-\frac{\sqrt{5}}{10}\right)(2-\sqrt{5})^n\]

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## More answers

- Zarkon

Looks like I'm a little off.

- Zarkon

I think I see my error

- Zarkon

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

- Zarkon

messed up my calculations that is :)

- asnaseer

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

- Zarkon

and I used your result to get mine :)

- asnaseer

:)

- asnaseer

Zarkon - I am convinced that your brain lives in another parale universe! :D

- Zarkon

could be ;)

- asnaseer

Wow! I just checked Zarkon's result and it is actually correct - not that I had any doubt of course ;-)

- Zarkon

lol

- Zarkon

there are techniques to solve these kinds of problems.

- asnaseer

would you be able to give any helpful pointers to the types of topics to study for these problems?

- Zarkon

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

- asnaseer

how do you leap from the first equation to the second? the first one involves terms in n, n-1 and n-2?

- Zarkon

Look at this ...
http://en.wikipedia.org/wiki/Recurrence_relation#Solving

- asnaseer

does this come under "Number Theory"?

- asnaseer

oh - ok - thanks for the link - I love learning new things! :)

- Zarkon

they are related to differential equations

- asnaseer

really - that is very interesting. thanks again Zarkon for letting us "peek" a little inside your brain. :D

- Zarkon

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.)

- asnaseer

ok - I have plenty of reading material now. thanks again!
and thanks to Mr.Math for posing such a question!

- Mr.Math

Thanks Zarkon! You're the best. I will have a look at the link you posted above. And thanks to asnasser as well.

- Mr.Math

@Zarkon: I'm looking for good textbooks on PDE's, could you recommend one or two to me?

- Zarkon

I've never studied PDE's.
I've worked with ODE's and Stochastic differential equation but not PDE's

- Mr.Math

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?

- Zarkon

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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