Mr.Math
  • Mr.Math
Find \(a_n\) if \(a_0=1\) and \(a_{n+1}=2a_n+\sqrt{3a_{n}^2-2}\), \(n\ge 0\).
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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asnaseer
  • asnaseer
ok, first few terms come out as: 1, 3, 11, 41, 153
asnaseer
  • asnaseer
\[a_n=4a_{n-1}-a_{n-2}\]
Zarkon
  • 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
  • Zarkon
Looks like I'm a little off.
Zarkon
  • Zarkon
I think I see my error
Zarkon
  • 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
  • Zarkon
messed up my calculations that is :)
asnaseer
  • 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
  • Zarkon
and I used your result to get mine :)
asnaseer
  • asnaseer
:)
asnaseer
  • asnaseer
Zarkon - I am convinced that your brain lives in another parale universe! :D
Zarkon
  • Zarkon
could be ;)
asnaseer
  • asnaseer
Wow! I just checked Zarkon's result and it is actually correct - not that I had any doubt of course ;-)
Zarkon
  • Zarkon
lol
Zarkon
  • Zarkon
there are techniques to solve these kinds of problems.
asnaseer
  • asnaseer
would you be able to give any helpful pointers to the types of topics to study for these problems?
Zarkon
  • 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
  • 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
  • Zarkon
Look at this ... http://en.wikipedia.org/wiki/Recurrence_relation#Solving
asnaseer
  • asnaseer
does this come under "Number Theory"?
asnaseer
  • asnaseer
oh - ok - thanks for the link - I love learning new things! :)
Zarkon
  • Zarkon
they are related to differential equations
asnaseer
  • asnaseer
really - that is very interesting. thanks again Zarkon for letting us "peek" a little inside your brain. :D
Zarkon
  • 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
  • asnaseer
ok - I have plenty of reading material now. thanks again! and thanks to Mr.Math for posing such a question!
Mr.Math
  • 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
  • Mr.Math
@Zarkon: I'm looking for good textbooks on PDE's, could you recommend one or two to me?
Zarkon
  • Zarkon
I've never studied PDE's. I've worked with ODE's and Stochastic differential equation but not PDE's
Mr.Math
  • 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
  • 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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