$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove armada]{\displaystyle{#1}}\,}
\newcommand{\llaves}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\parcial #3^{#1}}}
\newcommand{\raíz}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\ds{f_{n} =
{8f_{n - 1} \over 5} + {6\raíz{4^{n - 1} - f_{n - 1}^{2}} \over 5}:\ {\large ?}\,,\qquad f_{0} = 0}$.
Permite a $\ds{f_{n} = 2^{n}\cos\pars{x_{n}}}$:
\begin{align}
2^{n}\cos\pars{x_{n}} & =
{8 \over 5}\,2^{n - 1}\cos\pars{x_{n - 1}} +
{6\root{2^{2n - 2} - 2^{2n - 2}\cos^{2}\pars{x_{n - 1}}} \over 5}
\\[5mm] & =
{8 \over 5}\,2^{n - 1}\bracks{\cos\pars{x_{n - 1}} +
{3 \over 4}\verts{\sin\pars{x_{n - 1}}}}\quad
\pars{\begin{array}{l}
\mbox{Note that the correct term is}
\\
\ds{\color{red}{\verts{\sin\pars{x_{n - 1}}}}}\,\,\, \mbox{instead of}
\\
\ds{\sin\pars{x_{n - 1}}}\ \mbox{since}
\\
\ds{\left.\root{a^{2}}\right\vert_{\ a\ \in\ \mathbb{R}} = \verts{a}}.
\end{array}}
\\[5mm] & =
{8 \sobre 5}\,2^{n - 1}\bracks{\cos\pars{x_{n - 1}} +
\tan\pars{\theta}\verts{\sin\pars{x_{n - 1}}}}
\end{align}
donde$\ds{\theta = \arctan\pars{3 \over 4}}$$\ds{x_{0} = \pi/2}$.
\begin{align}
\cos\pars{x_{n}} & =
{4 \over 5}\,\sec\pars{\theta}\cos\pars{x_{n - 1} -
\mrm{sign}\pars{\sin\pars{x_{n - 1}}}\theta}
\\[5mm] & =
{4 \over 5}\,\root{\pars{3/4}^{2} + 1}
\cos\pars{x_{n - 1} - \mrm{sign}\pars{\sin\pars{x_{n - 1}}}\theta}
\\[5mm] \implies &
\bbx{\cos\pars{x_{n}} = \cos\pars{x_{n - 1} -
\mrm{sign}\pars{\sin\pars{x_{n - 1}}}\theta}}
\end{align}
Por otra parte,
\begin{align}
x_{n} & = x_{n - 1} -
\mrm{sign}\pars{\sin\pars{x_{n - 1}}}\theta
\\[5mm]
\sum_{k = 1}^{n}x_{k} & = \sum_{k = 1}^{n}x_{k - 1} -
\theta\sum_{k = 1}^{n}\mrm{sign}\pars{\sin\pars{x_{k - 1}}}
\\[5mm]
-\,{\pi \over 2} + \sum_{k = 0}^{n}x_{k} & = \sum_{k = 0}^{n - 1}x_{k} -
\theta\sum_{k = 1}^{n}\mrm{sign}\pars{\sin\pars{x_{k - 1}}} =
\sum_{k = 0}^{n - 1}x_{k} -
\theta\sum_{k = 0}^{n - 1}\mrm{sign}\pars{\sin\pars{x_{k}}}
\\[5mm]
\implies &
\bbx{x_{n} =
{\pi \over 2} -
\bracks{\sum_{k = 0}^{n - 1}\mrm{sign}\pars{\sin\pars{x_{k}}}}\theta\,,
\qquad x_{0} = {\pi \over 2}\,,\quad\theta = \arctan\pars{3 \over 4}}
\end{align}
$$
\mbox{Entonces}\quad x_{0} = {\pi \over 2}\,,\ x_{1} = {\pi \over 2} - \theta\quad \mbox{y}\quad
\left\{\begin{array}{rcl}
\ds{x_{2}} & \ds{=} & \ds{{\pi \over 2} - 2\theta}
\\[1mm]
\ds{x_{3}} & \ds{=} & \ds{{\pi \over 2} - 3\theta}
\\[1mm]
\ds{x_{4}} & \ds{=} & \ds{{\pi \over 2} - 2\theta}
\\[1mm]
\ds{x_{5}} & \ds{=} & \ds{{\pi \over 2} - 3\theta}
\\[1mm]
\ds{\vdots} & \ds{\vdots} & \ds{\phantom{AA}\vdots}
\\[1mm]
\ds{x_{10}} & \ds{=} & \ds{{\pi \over 2} - 2\theta}
\end{array}\right.
$$
$$
f_{10} = 2^{10}\
\underbrace{\cos\pars{{\pi \over 2} - 2\arctan\pars{3 \más de 4}}}
_{\ds{=\ {24 \más de 25}}} = \bbx{24576 \más de 25} = 983.04
$$
