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\begin{document}
\begin{center}
{\LARGE Week 2, Due Fri 10/11}
\end{center}


\begin{enumerate}


\item Spivak, Chapter 2, Problems: $3, 12, 19, 26$


\item Let~$p_n = 1,1,3,7,17 \ldots$ and~$q_n = 0,1,2,5,12, \ldots$ be sequences
defined by the initial values~$p_0=1, p_1=1$ and~$q_0=0$, $q_1=1$ and then
\[\begin{aligned}
p_{n} & = 2 p_{n-1} + p_{n-2}, \\
q_{n} & = 2 q_{n-1} + q_{n-2}.\end{aligned} 
\]
Prove by induction that
\[ p^2_n - 2 q^2_n = (-1)^n.\]

\item Let \(b > a\) be non-negative integers.
Let
\[H_{a,b} =  \frac{1}{a+1} + \frac{1}{a+2} + \ldots + \frac{1}{b-1} + \frac{1}{b}\]
be the sums of the reciprocals of all integers between \(a+1\) and \(b\), and let
\[H_n = H_{0,n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}.\]
\begin{enumerate}
\item Prove that
\(\displaystyle{H_{a,b} \ge \frac{b-a}{b}.}\)
\item Prove by induction that
\(\displaystyle{H_{2^n} \ge 1 + n \cdot \frac{1}{2}}\) for all \(n \in \N\).
\end{enumerate}
\end{enumerate}







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