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%%% file:      s5.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2016/12/06                                                    %%%
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\begin{document}

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\title{5. domáca úloha z predmetu 1-AIN-160 Matematika (3) ZS 2016/17}

\author{Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}

Vyriešte nasledujúcu rekurentnú rovnicu:
\begin{align*}
a_0 & = 0
\\
a_1 & = 3
\\
a_2 & = 11
\\
a_{n+3} - 5 a_{n+2} + 8 a_{n+1} - 4 a_n & = 0
.
\end{align*}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Uvažujme slová vytvorené zo znakov abecedy $\{ a,b,c,d,e,f \}$.
Nech $b_n$ je počet slov dĺžky $n \geq 0$,
ktoré neobsahujú dve po sebe idúce spoluhlásky.
Nájdite rekurentný vzťah pre postupnosť $b_n$.
Uveďte vhodné počiatočné podmienky.
Rekurentnú rovnicu vyriešte.

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\section*{3. príklad}

Vyriešte nasledujúcu rekurentnú rovnicu:
\begin{align*}
c_0 & = 1
\\
c_{n+1} - 2 c_n & = 2^{n+1} + 3^n
.
\end{align*}

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\subsection*{Riešenie 3. príkladu}

Táto časť obsahuje riešenie 3. príkladu.

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\end{document}

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