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I have the sequence defined by,

$a_{n+2} = \frac{1}{3} a_{n+1} + \frac{2}{3}a_n$,

where $a_0 = 0, a_1 = 1$.

I have proved that this converges, but I cannot figure out what it converges to. I tried to find some pattern in the terms so that I could find a closed-form of each term, but I had no luck.

Souroy
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  • One possibility is to transform this into a matrix and diagonalize it. This will give you the $n^{th}$ term of the recursion. – Marcos Feb 23 '22 at 11:07
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    Note that $a_{n+2}-a_{n+1}=-\frac{2}{3}(a_{n+1}-a_n)$, so the sequence $a_{n+1}-a_n$ is a geometric progression. – Joshua Woo Feb 23 '22 at 11:07

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