How to use three dots and brackets with TikZ to draw Automaton state machine?

Question

I tried to add a curly braces below all the states in order to group them, but I couldn't find a way to make it work. Furthermore, how could I draw a three dots as a state since I need to prove a theorem by induction on n. Here is what I have so far.

\documentclass[10pt,letterpaper]{article}
\usepackage[latin1]{inputenc}
\usepackage[left=1in,right=1in,top=1in,bottom=1in]{geometry} 
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{automata,positioning}

\begin{document}
    \begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto] 
        \node[state,initial]                (q_0)                           {$a$};
        \node[state]                        (q_1)   [right=of q_0]          {$a^2$};
        \node[state]                        (q_2)   [right=of q_1]          {$a^3$};
        \node[state,accepting]              (q_3)   [right=of q_2]          {$a^n$};    

        \path[->]
        (q_0)   edge                        node {a}            (q_1)  
        (q_1)   edge                        node {a}            (q_2) 
        (q_2)   edge                        node {...}          (q_3) 
        (q_3)   edge    [loop above]        node {a}            (q_3)
        ; %end path 
    \end{tikzpicture}   \\
\end{document}  

This is the picture that illustrates what I meant:

Thank you

Answer 1

\documentclass[10pt,letterpaper]{article}
\usepackage[latin1]{inputenc}
\usepackage[left=1in,right=1in,top=1in,bottom=1in]{geometry} 
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,automata,calc,positioning}

\begin{document}
    \begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto] 
        \node[state,initial]                (q_0)                           {$a$};
        \node[state]                        (q_1)   [right=of q_0]          {$a^2$};
        \node[state]                        (q_2)   [right=of q_1]          {$a^3$};
        \node[state,accepting]              (q_3)   [right=of q_2]          {$a^n$};    

        \path[->]
        (q_0)   edge                        node {a}            (q_1)  
        (q_1)   edge                        node {a}            (q_2) 
        (q_2)   edge                        node {...}          (q_3) 
        (q_3)   edge    [loop above]        node {a}            (q_3)
        ; %end path 
\draw[decorate,decoration={brace,mirror,raise=6pt}, thick] ($(q_0.south west)+(-1.5,0)$)--(q_3.south east);  
    \end{tikzpicture}  

\end{document} 

Answer 2

Just a little change over Altermundus' answer with \dots being an state.

\documentclass[10pt,letterpaper]{article}
\usepackage[latin1]{inputenc}
\usepackage[left=1in,right=1in,top=1in,bottom=1in]{geometry} 
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{automata,positioning,decorations.pathreplacing}

\begin{document}
    \begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid,auto] 
    \node[state,initial] (q_0) {$a$};
    \node[state] (q_1) [right=of q_0] {$a^2$};
    \node[state] (q_2) [right=of q_1] {$a^3$};
      \node        (q_dots) [right=of q_2] {$\cdots$}; 
    \node[state,accepting] (q_3) [right=of q_dots] {$a^n$};    

    \path[->]
    (q_0) edge node {a} (q_1)  
    (q_1) edge node {a} (q_2) 
    (q_2) edge  node {a} (q_dots) 
    (q_dots) edge node{a} (q_3)
    (q_3) edge [loop above] node {a} (q_3)
    ; %end path 

\draw [decorate,decoration={brace,amplitude=10pt,mirror,raise=10pt},yshift=0pt]
(q_0.south west) -- (q_3.south east);

\end{tikzpicture}   \\
\end{document}