%
% Sigma Detectives
% Activity 2
% (sigma notation - sec. 5.1 and 5.2)
% An activity borrowed from Harel Barzilai who extended the activity in
%     Cooperative learning in Mathematics: a handbook for teachers
%     edited by Neil Davidson
%

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\begin{center}
\large\bf Project Based Math 112, Fall 2001\\
Activity \#2 --- Sigma Detectives\\[.25in]
\end{center}

Fill in the following chart.  First, calculate the sums for specific
values of $n$.  Then, make a conjecture (educated guess) about what
{\it formula} gives the value of each sum for arbitrary $n$.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& $n=1$ & $n=2$ & $n=3$ & $n=4$ & $n=5$ & Conjectured formula\\\hline\hline
$\displaystyle\sum_{i=1}^n 2i$ &&&&&&\\\hline
$\displaystyle\sum_{i=1}^n i$ &&&&&&\\\hline
$\displaystyle\sum_{i=1}^n i^3$ &&&&&&\\\hline
\end{tabular}
\end{center}

Scratchwork:

\vfill

\begin{center}
{\sc Over} $\Longrightarrow$
\end{center}
\newpage

{\bf Challenge:} Prove the formula for $\sum_{i=1}^n 2i$
(and thus for $\sum_{i=1}^n i$).

{\bf Suggestion:} Rewrite the first sum as the sum of two rows of numbers:

\begin{center}
$\begin{array}{rr@{\ +\ }c@{\ +\ }c@{\ +\ }c@{\ +\ }c@{\ +\ }l}
  & 1 & 2 & 3 & \cdots & n\!-\!1 & n\\
+ & n & n\!-\!1 & n\!-\!2 & \cdots & 2 & 1\\\hline
=
\end{array}$
\end{center}

What do you notice as you sum these numbers?
(The great mathematician Karl Friedrich Gauss (1777-1855) discovered
this ``trick'' in grade school!)

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