% You should title the file with a .tex extension (hw1.tex, for example) \documentclass[11pt]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage{fancyhdr} \oddsidemargin0cm \topmargin-2cm %I recommend adding these three lines to increase the \textwidth16.5cm %amount of usable space on the page (and save trees) \textheight23.5cm \newcommand{\question}[2] {\vspace{.25in} \hrule\vspace{0.5em} \noindent{\bf #1: #2} \vspace{0.5em} \hrule \vspace{.10in}} \renewcommand{\part}[1] {\vspace{.10in} {\bf (#1)}} \newcommand{\myname}{Write Yr Name here!} \newcommand{\myandrew}{write-yr-andrew-id-here@andrew.cmu.edu} \newcommand{\myhwnum}{homework-number-here} \pagestyle{fancyplain} \lhead{\fancyplain{}{\textbf{HW\myhwnum}}} % Note the different brackets! \rhead{\fancyplain{}{\myname\\ \myandrew}} \chead{\fancyplain{}{15-251}} \begin{document} \medskip % Skip a "medium" amount of space % (latex determines what medium is) % Also try: \bigskip, \littleskip \thispagestyle{plain} \begin{center} % Center the following lines {\Large 15-251 Assignment \myhwnum} \\ \myname \\ \myandrew \\ Collaborators: no one / hbovik,johnq,fhacker\\ Your recitation section \\ The date \\ \end{center} \question{1}{Mathematical Symbols} This is an example of an answer to a homework question. In your answer, you may want to use a variety of mathematical symbols: % this is one way to make a list. Another would be to say % \begin{enumerate} and \end{enumerate}, which would give numbers instead % of bullets to the items \begin{itemize} \item Fractions: $\frac{2}{3}$, or (equivalently) ${2 \over 3}$ \item Binomial coefficients: ${5 \choose 2} = 10$ \item Subscripts and superscripts: $t_0$, $t^2$, $t_0^{2 \over 3}$, \item Greek letters: $\alpha$, $\beta$, $\gamma$, $\lambda$, $\Pi$. \item Summations: $\sum_{i=1}^n i = {n(n+1) \over 2}$. \end{itemize} You can refer to Leslie Lamport's ``\LaTeX\ User's Guide and Reference Manual'' for more useful info on mathematical typesetting with \LaTeX. Pages 42-46 outline many of the useful math symbols and functions. Another good reference for the mathematical symbols of \LaTeX\ is ``Inessential LaTeX'' by the MIT SIPB group. Find it on the Web at \verb|http://www.mit.edu/afs/sipb/project/doc/latex/guide.PS| \question{2}{Little Gauss's Formula} This is another example of a question. In this case, it's a multi-part question. \part{a} Recall {\em Little Gauss's formula}: % equations are automatically centered! \begin{eqnarray} \sum_{i=1}^n i = {{n(n+1)} \over 2} % as usual, liberal use of curly braces... \label{little-gauss} % labelled so we can refer to this formula by number \end{eqnarray} \part{b} Now, equation \ref{little-gauss} can be proven by induction as follows: \begin{itemize} \item {\bf Base case}: $n=1$: $1 = 1(2)/2=1$. \item {\bf Inductive hypothesis}: assume the equation holds for $n=2...k$. \item {\bf Inductive step}: for $n=k+1$, we have \begin{eqnarray*} % the star suppresses the equation numbers \sum_{i=1}^{k+1} i = (k+1) + \sum_{i=1}^k i \end{eqnarray*} Using the inductive hypothesis, we can substitute for the second term on the righthand side: \begin{eqnarray*} \sum_{i=1}^{k+1} i &=& (k+1) + k(k+1)/2\\ &=& {{2k+2 + k(k+1)} \over 2}\\ &=& {{k^2 + 3k + 2} \over 2}\\ &=& {{(k+1)(k+2)} \over 2} \end{eqnarray*} % notice \\ to indicate newline and the &=& to line up the equals signs \end{itemize} Lo and behold! The last line shows that for $n=k+1$, little Gauss' formula still holds for $n=k+1$! We've showed that the formula holds for $n=1$, and we've shown that if it holds for $n=k$ it must hold for $n=k+1$. Therefore, it must hold for all $n$. \end{document}