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\lhead{MATH 387-01}
\chead{Problem Set \#6\answer{ solutions}}
\rhead{\qonly{Name: \framebox{\phantom{Your name goes here}}}}
\lfoot{}
\cfoot{Page \thepage\ of \pageref{LastPage}}
\rfoot{{\bf due November 23, 2015}}
\begin{document}
\begin{enumerate}
\item {\bf (10 points)} \question{You open an account at a bank that
pays 5\% interest yearly, and deposit $a_0$ dollars in it. Every
year you withdraw \$10 times the number of years you have had the
account. For example, if you started with \$1000, then in the
first year you would earn \$50 in interest and withdraw \$10,
leaving \$1040, and in the second year would earn \$52 and
withdraw \$20, leaving \$1072, and so forth.
\begin{enumerate}
\item \question{Find a recurrence for $a_n$, the balance in the
account after $n$ years.}
\answer{}
\item \question{Solve the recurrence to find a closed form for
$a_n$.}
\answer{}
\item \question{What is the smallest initial deposit which would
guarantee that the account never runs out of money?}
\answer{}
\end{enumerate}
}
\item {\bf (10 points)} \question{Let $a_n=8a_{n-1}-16a_{n-2}+3\cdot
4^n$ with $a_0=3$ and $a_1=-1$. Find a closed form for $a_n$.}
\answer{}
\item {\bf (10 points)} \question{We are making bracelets with 6
stones in a ring, with three different colors of stone. A bracelet
must contain at least one stone of each color. Two bracelets are
considered to be identical if one is simply a rotation or a flip
of the other. How many different bracelets are possible?}
\answer{}
\item {\bf (10 points)} \question{A $4\times 4$ grid of squares is
filled in, with each of the 16 squares colored black or white. Two
colorings are regarded as identical if one can be converted to
each other by performing any combination of flipping, rotating, or
swapping the two colors (flipping all the black squares to white
and vice versa). How many non-identical colorings are there?}
\answer{}
\end{enumerate}
\vfill
\begin{center}
\fbox{
\begin{minipage}{6in}
Guided only by their feeling for symmetry, simplicity, and
generality, and an indefinable sense of the fitness of things,
creative mathematicians now, as in the past, are inspired by the
art of mathematics rather than by any prospect of ultimate
usefulness.\\
\phantom0\hfill ---Eric Temple Bell
\end{minipage}
}
\end{center}
\end{document}