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

1. asnaseer

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

2. asnaseer

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

3. Zarkon

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

4. Zarkon

Looks like I'm a little off.

5. Zarkon

I think I see my error

6. Zarkon

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

7. Zarkon

messed up my calculations that is :)

8. 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.

9. Zarkon

and I used your result to get mine :)

10. asnaseer

:)

11. asnaseer

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

12. Zarkon

could be ;)

13. asnaseer

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

14. Zarkon

lol

15. Zarkon

there are techniques to solve these kinds of problems.

16. asnaseer

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

17. 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

18. asnaseer

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

19. Zarkon

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

20. asnaseer

does this come under "Number Theory"?

21. asnaseer

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

22. Zarkon

they are related to differential equations

23. asnaseer

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

24. 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.)

25. asnaseer

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

26. 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.

27. Mr.Math

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

28. Zarkon

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

29. 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?

30. 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.