document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1046510>Question 1046510</A>: <I><SPAN STYLE=\"word-break: break-all;\"> `Take a look at the following sums:\r<BR>\n" );
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document.write( "1=1<BR>\n" );
document.write( "1+3=4<BR>\n" );
document.write( "1+3+5=9<BR>\n" );
document.write( "1+3+5+7=16<BR>\n" );
document.write( "1+3+5+7+9=25\r<BR>\n" );
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document.write( "a.  Come up with a conjecture about the sum when you add the first n odd numbers.  For example when you added the first five odd numbers ( 1+3+5+7+9) what do you get?  What if I wanted to add the first 10 odd numbers? Or 100? \r<BR>\n" );
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document.write( "b.  Can you think of a geometric interpretation of this pattern? If you start with one square and add on three more, what can you make? If you now have four squares and you add on five more, what can you make? \r<BR>\n" );
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document.write( "c.  Is there a similar pattern for adding the first n even numbers? \r<BR>\n" );
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document.write( "2=2<BR>\n" );
document.write( "2+4=6<BR>\n" );
document.write( "2+4+6=12<BR>\n" );
document.write( "2+4+6+8= 20<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #661977 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=661977\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "The sum of the first <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> odd numbers is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n^2\">.\r<BR>\n" );
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document.write( "Induction Proof:\r<BR>\n" );
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document.write( "Prove:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\forall\ n, k\ \in\ \mathbb{N}\right)\ \sum_{k\,=\,1}^n\,2k\,-\,1\ =\ n^2\">\r<BR>\n" );
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document.write( "Initial Step:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n\ =\ 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sum_{k\,=\,1}^1\ 2n\ -\ 1\ =\ 2(1)\ -\ 1\ =\ 1\ =\ 1^2\">\r<BR>\n" );
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document.write( "Induction Step:<BR>\n" );
document.write( "Show:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\sum_{k\,=\,1}^n\,2k\,-\,1\ =\ n^2\right)\ \right\ \sum_{k\,=\,1}^{n\,+\,1}\,2k\,-\,1\ =\ (n\ +\ 1)^2\">\r<BR>\n" );
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document.write( "Assume:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sum_{k\,=\,1}^n\,2k\,-\,1\ =\ n^2\">\r<BR>\n" );
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document.write( "Then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2(n\ +\ 1)\ -\ 1\ =\ 2n\ +\ 1\">\r<BR>\n" );
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document.write( "so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sum_{k\,=\,1}^{n\,+\,1}\,2k\,-\,1\ =\ \sum_{k\,=\,1}^n\,2k\,-\,1\ +\ 2n\ +\ 1\">\r<BR>\n" );
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document.write( "Substitute from the assumption:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sum_{k\,=\,1}^{n\,+\,1}\,2k\,-\,1\ =\ n^2\  +\ 2n\ +\ 1\">\r<BR>\n" );
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document.write( "But\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ n^2\ +\ 2n\ +\ 1\ =\ (n\ +\ 1)^2\">\r<BR>\n" );
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document.write( "Therefore, if the assertion is true for some <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\">, then it is true for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\ +\ 1\">.  QED.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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