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Math 192r, Problem Set \#1 \\
(due 9/20/01)
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\begin{enumerate}
\item
\begin{enumerate}
\item Write and run a program to compute
$f(n) = \sum_{k=0}^n (-1)^k {n \choose k}^2$.
(Submit this part by email.)
\item Devise a conjecture about the value of $f(n)$.
\item Prove your conjecture using the algebraic interpretation of $n \choose k$
as the coefficient of $x^k$ in $(1+x)^n$.
\item Prove your conjecture using the interpretation of $n \choose k$
as the number of combinations of $n$ things taken $k$ at a time.
\end{enumerate}
\item
Define a sequence of functions $f_0(x), f_1(x), f_2(x), ...$
where $f_0(x) = x$, $f_1(x) = x$,
and for all $n>1$, $f_n(x) = ([f_{n-1}(x)]^2+1)/f_{n-2}(x)$.
Thus, $f_2(x) = x + x^{-1}$, $f_3(x) = x + 3x^{-1} + x^{-3}$, etc.
\begin{enumerate}
\item
Formulate a conjecture about the values of $f_n(1)$.
\item
Formulate a conjecture about the values of $f_n(-1)$.
\item
Formulate a conjecture about the values of $f_n(i)$,
where $i = \sqrt{-1}$.
\item
Formulate a conjecture about the coefficients
of the polynomials $f_n(x)$.
\end{enumerate}
\end{enumerate}
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In each case, if you can't get all the way through, explain how far you got and what the obstacles were.
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\noindent
Please be sure to write down how many hours you spent working on the assignment,
and whom you worked with (something you should do on ALL your assignments for
this course).
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