Inserting a series of equally-spaced dots in math-mode

Question

1. A series of equally-spaced dots are required to complete the set up.

2. A slight adjustment of the vertical alignment is required.

The above set up must appear as something like this:-

\documentclass[11pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{booktabs}

\begin{document}
using the detached coefficients, we have the scheme\\\\
\noindent
$\begin{array}{@{} >{{}}r @{} >{{}}r *{20}{ @{}>{{}}r<{{}}@{} } }
& 1 & + & _nC_{1} & + & _nC_{2} & + & _nC_{3} & + & \dotsb  & + & _nC_{n-1} & + & _nC_{n} \\
\times \\
& 1 & + &  1      & \\
\midrule
& 1 &   &         & + & _nC_{1} &   &         & + & _nC_{2} &   &   &   &   &   &   &         & + & _nC_{n} \\
&   &   &         & + &  1      &   &         & + & _nC_{1} &   &   &   &   &   &   &      & + & _nC_{n-1} & + & _nC_{n} \\
\midrule
\end{array}$
\end{document}

Answer 1

You can use \hdotsfor from amsmath (the argument indicates the number of columns to be spanned):

\documentclass[11pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{booktabs}

\begin{document}
using the detached coefficients, we have the scheme\\\\
\noindent
$\begin{array}{@{} >{{}}r @{} >{{}}r *{20}{ @{}>{{}}r<{{}}@{} } }
& 1 & + & _nC_{1} & + & _nC_{2} & + & _nC_{3} & + & \dotsb  & + & _nC_{n-1} & + & _nC_{n} \\
\times \\
& 1 & + &  1      & \\
\midrule
& 1 &   &         & + & _nC_{1} & \hdotsfor{2}          & + & _nC_{2} &   \hdotsfor{3}           + & _nC_{n} \\
&   &   &         & + &  1  & \hdotsfor{2}         & + & _nC_{1} &  \hdotsfor{3}       + & _nC_{n-1} & + & _nC_{n} \\
\midrule
\end{array}$
\end{document}

With l (left) aligned columns strating from the second one:

\documentclass[11pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{booktabs}

\begin{document}
using the detached coefficients, we have the scheme\\\\
\noindent
$\begin{array}{@{} >{{}}r @{} >{{}}r *{20}{ @{}>{{}}l<{{}}@{} } }
& 1 & + & _nC_{1} & + & _nC_{2} & + & _nC_{3} & + & \dotsb  & + & _nC_{n-1} & + & _nC_{n} \\
\times \\
& 1 & + &  1      & \\
\midrule
& 1 &   &         & + & _nC_{1} & \hdotsfor{2}          & + & _nC_{2} &   \hdotsfor{3}           + & _nC_{n} \\
&   &   &         & + &  1  & \hdotsfor{2}         & + & _nC_{1} &  \hdotsfor{3}       + & _nC_{n-1} & + & _nC_{n} \\
\midrule
\end{array}$
\end{document}