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\documentclass{math215}

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\usepackage{times,txfonts}

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\doclabel{Math 215: Homework 8a}
\docauthor{Your name here}
\docdate{March 22, 2013}

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% by uncommenting the line below.  Feel free to make changes to that file.
\input{math215extras.tex}

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\begin{document}


\begin{proposition}{4.8} For all $k\in \Nats$, $4^k>k$.
\end{proposition}
\begin{pf} 
Your proof goes here.
\end{pf}

\begin{proposition}{4.13} For $x\neq 1$ and $k\in\Ints_{\ge 0}$,
$\displaystyle \sum_{j=0}^k x^j = \frac{1-x^{k+1}}{1-x}$.
\end{proposition}
Hint: Show that $(1-x)\sum_{j=0}^k x^j = 1-x^{k+1}$.  
\begin{pf} 
Your proof goes here.
\end{pf}

\begin{proposition}{4.15(i)} Let $m\in\Ints$ and $(x_j)_{j=1}^\infty$
  be a sequence in $\Ints$.  If 
  then for all $k\in\Nats$
\[
\sum_{j=1}^k m x_j = m \sum_{j=1}^k x_j.
\]
\end{proposition}
\begin{pf} 
Your proof goes here.
\end{pf}

\begin{proposition}{4.15(iii)} Let $(x_j)_{j=1}^\infty$
  be a sequence in $\Ints$.  If $x_j=n\in\Ints$ for all $j\in\Nats$ 
  then for all $k\in\Nats$
\[
\sum_{j=1}^k x_j = kn.
\]
\end{proposition}
\begin{pf} 
Your proof goes here.
\end{pf}

\begin{proposition}{4.16(ii)}  Let $(x_j)_{j=m}^\infty$ and $(y_j)_{j=m}^\infty$ be sequences in $\Ints$.  For all $a,b\in\Ints$
such that $m\le a \le b$, 
\[
\sum_{j=a}^b (x_j+y_j) = \sum_{j=a}^b x_j +\sum_{j=a}^b  y_j. 
\]
\end{proposition}
\begin{pf} 
Your proof goes here.
\end{pf}

\begin{proposition}{4.18}  Let $(x_j)_{j=1}^\infty$ and $(y_j)_{j=1}^\infty$ be sequences in $\Ints$ such that $x_j\le y_j$ for all $j\in\Nats$.  
Then for all $k\in\Nats$,
\[
\sum_{j=1}^k x_j \le \sum_{j=1}^k y_j.
\]
\end{proposition}
\begin{pf} 
Your proof goes here.
\end{pf}

\begin{proposition}{4.30} For all $k,m\in\Nats$, where $m\ge 2$, 
\[
f_{m+k} = f_{m-1} f_k + f_m f_{k+1}.
\]
\end{proposition}
\begin{pf} 
Your proof goes here.
\end{pf}

\end{document}