My Math Forum limit of a recursive sequence

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 May 9th, 2015, 05:14 AM #1 Member   Joined: Oct 2013 Posts: 36 Thanks: 0 limit of a recursive sequence let (a_n) be a sequence such that: a_(n+2)=1/3*a_(n+1)+2/3*a_n what is the limit of (a_n)? [the fact it converges has already been proved earlier in the question and we are not given a(1) and a(2)]
 May 9th, 2015, 05:22 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,634 Thanks: 2620 Math Focus: Mainly analysis and algebra The general solution is$$A\lambda_1^n + B\lambda_2^n$$ where $\lambda_1$ and $\lambda_2$ are roots of the characteristic equation $u^2 - \frac13u - \frac23=0$. You probably don't need to know the constants $A$ and $B$. Thanks from fromage
 May 9th, 2015, 06:55 AM #3 Member   Joined: Oct 2013 Posts: 36 Thanks: 0 how did u get this? (a derivation would be greatly appreciated but a brief explanation would also be very helpful if u dont mind)
 May 9th, 2015, 07:10 AM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,634 Thanks: 2620 Math Focus: Mainly analysis and algebra I haven't seen a constructive proof, but a first order difference equation $a_{n+1} = ka_n$ has solution $a_0 k^n$ which could motivate us to seek solutions of the form $a_n = Ar^n$ to the second order equation $a_{n+2}+ba_{n+1}+ca_n=0$. Substituting that into the original equation and solving for $r$ gives the result that $r$ must be a root of the characteristic equation $u^2 + bu + c = 0$. Since this equation has two roots, there are two independent solutions and any linear combination of them will also be a solution because the equation is linear. One ought to prove that such a construction provides all solutions to the equation, but I've never seen such a proof.
 May 9th, 2015, 07:21 AM #5 Member   Joined: Oct 2013 Posts: 36 Thanks: 0 the thing is this is from an analysis paper (the syllabus is a bit older but we have definitely not touched on difference equations yet, thats in another course altogether) so is there another way you could see to derive this using only 1st year analysis techniques?
 May 9th, 2015, 09:33 AM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,634 Thanks: 2620 Math Focus: Mainly analysis and algebra It seems an odd question to give to someone who hasn't studied the subject at all!
May 9th, 2015, 09:41 AM   #7
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
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Hello, fromage!

Quote:
 $$$a_n$$\text{ is a sequence such that: }\:a_{n+2}\:=\:\frac{1}{3}a_{n+1}\,+\,\frac{2}{3}a _n$ $\tex{What is the limit of }(a_n)\,?$

$\text{Let: }\,a_1= p,\;a_2 = q.$

$\text{Let }\,a_n \,=\,X^n$
$\text{The equation becomes: }\:X^{n+2} \:=\:\frac{1}{3}X^{n+1}\,+\,\frac{2}{3}X^n$

$\text{W\!e have: }\:3X^{n+2} \,-\,X^{n+1}\,-\,2X^n \:=\:0$

$\text{Divide by }X^n:\;\;3X^2\,-\,X\,-\,2\:=\:0 \;\;\;\Rightarrow\;\;\;(X\,-\,1)(3X\,+\,2) \:=\:0$

$\text{Hence: }\:X \:=\:1,\,-\frac{2}{3}$
$\text{Thus: }\:f(n) \;=\; 1^n\text{ or }\left(-\frac{2}{3}\right)^n$

$\text{Assume that }f(n)\text{ is a linear combination of these roots.}$
$\;\;\;\text{That is: }\;f(n) \;=\;A(1^n)\,+\,B\left(-\frac{2}{3}\right)^n$

$\text{From the first two terms, we have:}$
$\;\;\;\begin{Bmatrix}f(1) = p: & A \,-\,\frac{2}{3}B &=& p \\ \\ \\
f(2) = q: & A\,+\,\frac{4}{9}B &=& q \end{Bmaix}$

$\text{Solve the system: }\;\begin{Bmatrix} A=&\frac{1}{5}(2a\,+\,3b) \\ \\ B=&\frac{9}{10}(b\,-\,a) \end{Bmatrix}=$

$\text{Therefore: }\;f(n) \;=\;\frac{1}{5}(2a\,+\,3b)\,+\,\frac{9}{10}(b\,-\,a)\left(-\frac{2}{3}\right)^n$

$\text{Now }you\text{ can find: }\:\lim_{n\to\infty} f(n)$

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