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

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

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

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

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

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.3ok...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
 4 years ago
Best ResponseYou've already chosen the best response.3messed up my calculations that is :)

asnaseer
 4 years ago
Best ResponseYou've already chosen the best response.1I arrived at my result by squaring the expression for \(a_{n+1}\) and also the expression for \(a_n\), then combining them both.

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

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

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

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

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

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.3take \[a_n=4a_{n1}a_{n2}\] 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

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

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

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

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

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.3they are related to differential equations

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

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.3I 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
 4 years ago
Best ResponseYou've already chosen the best response.1ok  I have plenty of reading material now. thanks again! and thanks to Mr.Math for posing such a question!

Mr.Math
 4 years ago
Best ResponseYou've already chosen the best response.0Thanks Zarkon! You're the best. I will have a look at the link you posted above. And thanks to asnasser as well.

Mr.Math
 4 years ago
Best 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?

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

Mr.Math
 4 years ago
Best ResponseYou've already chosen the best response.0Oh, 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
 4 years ago
Best ResponseYou've already chosen the best response.3Just 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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