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Mr.Math
Group Title
Find \(a_n\) if \(a_0=1\) and \(a_{n+1}=2a_n+\sqrt{3a_{n}^22}\), \(n\ge 0\).
 2 years ago
 2 years ago
Mr.Math Group Title
Find \(a_n\) if \(a_0=1\) and \(a_{n+1}=2a_n+\sqrt{3a_{n}^22}\), \(n\ge 0\).
 2 years ago
 2 years ago

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asnaseer Group TitleBest ResponseYou've already chosen the best response.1
ok, first few terms come out as: 1, 3, 11, 41, 153
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
\[a_n=4a_{n1}a_{n2}\]
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
\[a_n=\frac{\sqrt{5}+5}{10}(2+\sqrt{5})^n+\left(\frac{1}{2}\frac{\sqrt{5}}{10}\right)(2\sqrt{5})^n\]
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
Looks like I'm a little off.
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
I think I see my error
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
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

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
messed up my calculations that is :)
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
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

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
and I used your result to get mine :)
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
Zarkon  I am convinced that your brain lives in another parale universe! :D
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
Wow! I just checked Zarkon's result and it is actually correct  not that I had any doubt of course ;)
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
there are techniques to solve these kinds of problems.
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
would you be able to give any helpful pointers to the types of topics to study for these problems?
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
take \[a_n=4a_{n1}a_{n2}\] and write it as \[x^2=4x1\] \[x^24x+1\] find the roots of this simple quadratic. you will then have part of my answer above
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
how do you leap from the first equation to the second? the first one involves terms in n, n1 and n2?
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
Look at this ... http://en.wikipedia.org/wiki/Recurrence_relation#Solving
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
does this come under "Number Theory"?
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
oh  ok  thanks for the link  I love learning new things! :)
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
they are related to differential equations
 2 years ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
really  that is very interesting. thanks again Zarkon for letting us "peek" a little inside your brain. :D
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
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

asnaseer Group TitleBest ResponseYou've already chosen the best response.1
ok  I have plenty of reading material now. thanks again! and thanks to Mr.Math for posing such a question!
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
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

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
@Zarkon: I'm looking for good textbooks on PDE's, could you recommend one or two to me?
 2 years ago

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
I've never studied PDE's. I've worked with ODE's and Stochastic differential equation but not PDE's
 2 years ago

Mr.Math Group TitleBest ResponseYou've already chosen the best response.0
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

Zarkon Group TitleBest ResponseYou've already chosen the best response.3
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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