% (find-angg "LATEX/2010-2-MD-exercs-VR.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2010-2-MD-exercs-VR.tex && latex    2010-2-MD-exercs-VR.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2010-2-MD-exercs-VR.tex && pdflatex 2010-2-MD-exercs-VR.tex"))
% (eev "cd ~/LATEX/ && Scp 2010-2-MD-exercs-VR.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (defun d () (interactive) (find-dvipage "~/LATEX/2010-2-MD-exercs-VR.dvi"))
% (find-dvipage "~/LATEX/2010-2-MD-exercs-VR.dvi")
% (find-pspage  "~/LATEX/2010-2-MD-exercs-VR.ps")
% (find-pspage  "~/LATEX/2010-2-MD-exercs-VR.pdf")
% (find-xpdfpage "~/LATEX/2010-2-MD-exercs-VR.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvipdf         2010-2-MD-exercs-VR.dvi 2010-2-MD-exercs-VR.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2010-2-MD-exercs-VR.ps 2010-2-MD-exercs-VR.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 600 -P pk -o 2010-2-MD-exercs-VR.ps 2010-2-MD-exercs-VR.dvi && ps2pdf 2010-2-MD-exercs-VR.ps 2010-2-MD-exercs-VR.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage  "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2010-2-MD-exercs-VR.pdf" (ee-twupfile "LATEX/2010-2-MD-exercs-VR.pdf") 'over)
% (ee-cp "~/LATEX/2010-2-MD-exercs-VR.pdf" (ee-twusfile "LATEX/2010-2-MD-exercs-VR.pdf") 'over)
% (find-twusfile     "LATEX/" "2010-2-MD-exercs-VR")
% http://angg.twu.net/LATEX/2010-2-MD-exercs-VR.pdf

\documentclass[oneside]{book}
\usepackage[latin1]{inputenc}
\usepackage{edrx08}       % (find-dn4ex "edrx08.sty")
%L process "edrx08.sty"  -- (find-dn4ex "edrx08.sty")
\input edrxheadfoot.tex   % (find-dn4ex "edrxheadfoot.tex")
\begin{document}

\input 2010-2-MD-exercs-VR.dnt

%*
% (eedn4-51-bounded)

%Index of the slides:
%\msk
% To update the list of slides uncomment this line:
%\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")

\def\sumin{\sum_{k=1}^n}

Matemática Discreta

Exercícios de preparação para a VR

(Adaptados dos exercícios 3 e 4 da p.165 do Scheinerman)

Versão preliminar

\bsk

3a) Queremos provar que $\sumin (3k-2) = n(3n-1)/2$.

3b) Queremos provar que $\sumin k^3 = n^2 (n+1)^2/4$.

3c) Queremos provar que $\sumin 9·10^k = 10^n-1$.

3d) Queremos provar que $\sumin 1/(k(k+1)) = 1 - 1/(k+1)$.




%*

\end{document}

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