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Consider the following sequence:

$a_{n+2} = \frac{a_{n+1}+a_{n}}{2}$, $a_{1}$ and $a_{2}$ are given.

Write $a_{n}$ as a function of $a_{1}$ and $a_{2}$ and show that its limit is $\frac{a1 + 2a_{2}}{3}$

I think I am loosing myself on algebra here. I can't even do the first part. Any help is welcome, thanks a lot!

Raul Guarini Riva
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  • Write ehw characteristic equation of this difference relation: $2x^2-x-1=0$ and find its roots. Then the general term $a_n=...$ – Svetoslav Mar 17 '16 at 19:42
  • Ok, the roots are 1 and -1/2. But I can't see how that helps :/ I'm sorry. How could that help solving the first part, I mean, writing it as a function of a1 and a2? – Raul Guarini Riva Mar 17 '16 at 19:48

2 Answers2

1

$A = \pmatrix{1/2&1/2\\1&0}$

$\pmatrix{a_{n}\\a_{n-1}}=A\pmatrix{a_{n-1}\\a_{n-2}}$

$\pmatrix{a_{n+2}\\a_{n+1}}=A^n\pmatrix{a_2\\a_1}$

$A = PDP^{-1}; A^n =PD^nP^{-1}$

$A = \pmatrix{1&1\\1&-2}\pmatrix{1&0\\0&-1/2}\pmatrix{2/3&1/3\\1/3&-1/3}$

$A^n = \pmatrix{1&1\\1&-2}\pmatrix{1&0\\0&-2^{-n}}\pmatrix{2/3&1/3\\1/3&-1/3}$

Limit as $n\to \infty = \pmatrix{1&1\\1&-2}\pmatrix{1&0\\0&0}\pmatrix{2/3&1/3\\1/3&-1/3}$

$\pmatrix{2/3&1/3\\2/3&1/3}$

$\lim\limits_{n\to\infty} a_n = 2/3a_2 + 1/3 a_1$

Doug M
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The general term for this sequence is $a_{n} = (1)^{n} x + \left(-\frac{1}{2}\right)^{n} y$ where $x$ and $y$ are unknown. See that $1$ and $-\frac{1}{2}$ are roots of the characteristic equation. Then you have the system $$ \begin{array}{rcl} a_{1} & = & x - \frac{1}{2} y\\ a_{2} & = & x + \frac{1}{4} y. \end{array} $$ Solve for $x$ and $y$.

Imanol N.
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  • Alright, that helps a lot. But I need to do it without using difference equations. I'm writing the solutions of a problem set for a Calculus class that hasn't been presented to difference equations yet. Is there an workaround? – Raul Guarini Riva Mar 17 '16 at 20:03