\problemdata      
        {summations-2} % latex-friendly label for the prob.
        {Summations} % The topic(s) of the problem content.
        {?} % Source (if known)
        {S02 CP6W} % Usage (list ones you are aware of).
        {Christos Kapoutsis, S02} % Last revision info (author, date).



\begin{problem}
Using the method described in lecture, a truck can travel across any size
desert if there is a large enough supply of gas at the border of the
desert.  Show that if there is a large enough supply of gas at the border,
a truck can also make a \emph{round trip} across any size desert.

\solution{
Let $R_n$ be the distance a truck can travel into the desert \emph{and
return} on $n$ tanks of gas.  Clearly, $R_1 = 1/2$.

On $n+1$ tanks, the truck will travel distance $x$ and back $n$ times,
leaving $1-2x$ tanks of gas at distance $x$ into the desert on each trip.
It then makes one more one-way trip to $x$.  This leaves it with $n(1-2x)
+ 1-x$ tank of gas at position $x$.  Leaving an $x$th of a tank so it can
get back, if the remaining $(n(1-2x) + 1-x) -x = (n+1)(1-2x)$ tanks equal
$n$, it can execute the $n$-tank round trip strategy from position $x$ and
still return to the desert border.  So
\begin{eqnarray}
(n+1)(1-2x) & = & n\\
x & = & 1/2(n+1)\\
R_{n+1} & = & R_n + x = R_n + 1/2(n+1).
\end{eqnarray}
So
\[
R_n = 1/2 (1 + 1/2 + 1/3 + \cdots + 1/n) = H_{n}/2.
\]
}
\end{problem}

