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There is a recursive sequence defined by: $a_1=1$ and $a_{n+1} = a_n/(4\cdot a_n + 3)$. I've also been given this sequence: $b_n = (2\cdot a_n + 1)/a_n$ and I have found that $b_{n+1} = 3 \cdot b_n$. How can I find the explicit formula of $a$?

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    $\begingroup$ Well, $b_n=3^n$. $\endgroup$
    – plop
    Jun 3 '21 at 16:18
  • $\begingroup$ wow thanks! i did not think about that now its easy, thank you! $\endgroup$
    – S H
    Jun 3 '21 at 16:20
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In my answer to this question, I detailed the steps for solving a first-order rational difference equation such as $${ a_{n+1} = \frac{m\,a_n + x}{a_n + y} }$$

For your case $m=\frac 14$, $x=0$, $y=\frac 34$ makes the problem quite simple. Just follow the steps.

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  • $\begingroup$ yes thanks! the comment below my question solved my problem i just did not know how to close the question $\endgroup$
    – S H
    Jun 4 '21 at 9:29

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