$$\begin{gathered} \mathrm{interface}\left(\mathrm{rtablesize}\=25\right)\: \\ f\left[1\right]≔1\: \\ \mathbf{for} k \mathbf{from} 1 \mathbf{to} 12 \mathbf{do} \\ f_{k\+1}≔4\/\left(3\+2 f_k\right)\; \\ r\parallel \left(k\+1\right)≔\left[a_{k\+1}\,f_{k\+1}\,\mathrm{evalf}\left(f_{k\+1}\right)\right]\; \\ \mathbf{end} \mathbf{do}\: \\ \mathrm{Matrix}\(\[\left[a_1\,1\,1\right]\,r\parallel \left(2..12\right)\]\) \end{gathered}$$

Table 8.1.9(a)   Termwise generation of members of a sequence defined by a nonlinear recursion

Write the recursion equation

$q≔a\left(n\+1\right) \left(3\+2 a\left(n\right)\right)\=4$

$a\left(n\+1\right)\left(3\+2a\left(n\right)\right)\=4$

Apply the rsolve command, using the syntax shown, and assigning the solution to the name $A$ 

$A≔\mathrm{rsolve}\left(\left\{q\,a\left(1\right)\=1\right\}\,a\right)$

$\frac{1}{2}\frac{13{\left(\frac{3}{4}\+\frac{1}{4}\sqrt{41}\right)}^n\sqrt{41}-13{\left(\frac{3}{4}-\frac{1}{4}\sqrt{41}\right)}^n\sqrt{41}\+41{\left(\frac{3}{4}\+\frac{1}{4}\sqrt{41}\right)}^n\+41{\left(\frac{3}{4}-\frac{1}{4}\sqrt{41}\right)}^n}{5{\left(\frac{3}{4}\+\frac{1}{4}\sqrt{41}\right)}^n\sqrt{41}-5{\left(\frac{3}{4}-\frac{1}{4}\sqrt{41}\right)}^n\sqrt{41}\+41{\left(\frac{3}{4}\+\frac{1}{4}\sqrt{41}\right)}^n\+41{\left(\frac{3}{4}-\frac{1}{4}\sqrt{41}\right)}^n}$

Obtain $\underset{n\to \infty }{lim}{a}_n$ 

$\underset{n\to \infty }{lim}A$ = $\frac{13\sqrt{41}\+41}{10\sqrt{41}\+82}$$\stackrel{\text{at 10 digits}}{\to }$$0.8507810593$$$

Table 8.1.9(b)   Use of the rsolve command to obtain the explicit solution for $a_n$

$\mathrm{factor}\left(A\right)$

$-\frac{1}{32}\frac{\left(-3\+\sqrt{41}\right)\left(13{\left(\frac{3}{4}-\frac{1}{4}\sqrt{41}\right)}^n\sqrt{41}\+64{\left(\frac{3}{4}\+\frac{1}{4}\sqrt{41}\right)}^n-105{\left(\frac{3}{4}-\frac{1}{4}\sqrt{41}\right)}^n\right)}{5{\left(\frac{3}{4}-\frac{1}{4}\sqrt{41}\right)}^n\sqrt{41}-8{\left(\frac{3}{4}\+\frac{1}{4}\sqrt{41}\right)}^n-33{\left(\frac{3}{4}-\frac{1}{4}\sqrt{41}\right)}^n}$

Table 8.1.9(c)   Application of the factor command to the solution $A$

$\mathrm{seq}\left(\mathrm{simplify}\left(A\right)\,n\=1..12\right)$

$1\,\frac{4}{5}\,\frac{20}{23}\,\frac{92}{109}\,\frac{436}{511}\,\frac{2044}{2405}\,\frac{9620}{11303}\,\frac{45212}{53149}\,\frac{212596}{249871}\,\frac{999484}{1174805}\,\frac{4699220}{5523383}\,\frac{22093532}{25968589}$

Table 8.1.9(d)   Members of the sequence $\left\{a_n\right\}$ extracted from the explicit solution $A$