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%%% file:      s4.tex                                                        %%%
%%% author:    Ján Komara                                                    %%%
%%% email:     komara@fmph.uniba.sk                                          %%%
%%% copyright: public domain                                                 %%%
%%% version:   2017/11/26                                                    %%%
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\begin{document}

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\title{4. domáca úloha z predmetu 1-AIN-121 Matematika (1) ZS 2017/18}

\author{Ján Komara}

%\date{}
\date{\today}

\maketitle

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\section*{1. príklad}

Dokážte priamo pomocou matematickej indukcie, že pre každé prirodzené číslo $n$ 
platí
\begin{align*}
\sum_{i=0}^{n} \frac{2^{i+1}}{(-3)^i} = 
\frac{3^{n+1} - (-2)^{n+1}}{\frac{5}{2} \times 3^n}
.
\end{align*}

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\subsection*{Riešenie 1. príkladu}

Táto časť obsahuje riešenie 1. príkladu.

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\section*{2. príklad}

Nech $f_n$ je postupnosť prirodzených čísel definovaná vzťahom:
\begin{align*}
f_0 & = 0
\\
f_1 & = 1
\\
f_2 & = 2
\\
f_{n+3} & = f_{n+2} + 2 f_{n+1} + 3 f_n + 4
.
\end{align*}
Uvažujme ďalej funkciu $g$ definovanú primitívnou rekurziou so substitúciou
v parametroch:
\begin{align*}
g(0,a,b,c) & = a
\\
g(n+1,a,b,c) & = g(n,a+2b+3c+4,a,b)
.
\end{align*}
Dokážte, že pre každé prirodzené číslo $n$ platí
\begin{align*}
f_{n+2} = g(n,2,1,0)
.
\end{align*}

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\subsection*{Riešenie 2. príkladu}

Táto časť obsahuje riešenie 2. príkladu.

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\end{document}

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