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

Question

asnaseer · Answer

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

asnaseer · Answer

$$a_n=4a_{n-1}-a_{n-2}$$

zarkon · Answer

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

zarkon · Answer

Looks like I'm a little off.

zarkon · Answer

I think I see my error

zarkon · Answer

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 · Answer

messed up my calculations that is  :)

asnaseer · Answer

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 · Answer

and I used your result to get mine  :)

asnaseer · Answer

:)

asnaseer · Answer

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

zarkon · Answer

could be ;)

asnaseer · Answer

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

zarkon · Answer

lol

zarkon · Answer

there are techniques to solve these kinds of problems.

asnaseer · Answer

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

zarkon · Answer

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 · Answer

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

zarkon · Answer

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

asnaseer · Answer

does this come under "Number Theory"?

asnaseer · Answer

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

zarkon · Answer

they are related to differential equations

asnaseer · Answer

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

zarkon · Answer

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 · Answer

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

mr.math · Answer

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 · Answer

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

zarkon · Answer

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

mr.math · Answer

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 · Answer

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.