document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1045633>Question 1045633</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Using direct proof, prove that if n is a natural number, then n(n+1) is even. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #661112 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=661112\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i'm not real good at these, but let me give it a shot.\r<BR>\n" );
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document.write( "you want to prove that n * (n + 1) is even, when you are dealing with a natural number.\r<BR>\n" );
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document.write( "a natural number is all the positive integers i believe.\r<BR>\n" );
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document.write( "it does not include 0 as far as i remember.\r<BR>\n" );
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document.write( "you have 2 possibilities.\r<BR>\n" );
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document.write( "if n is even, then n + 1 is odd.\r<BR>\n" );
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document.write( "if no is odd, then n + 1 is even.\r<BR>\n" );
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document.write( "assuming that n is even, then there exists a number k such that 2 * k = n.\r<BR>\n" );
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document.write( "n * (n + 1) becomes 2 * k * (2 * k + 1) which becomes 4 * k^2 + 2 * k.\r<BR>\n" );
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document.write( "the number 4 * k^2 + 2 * k has to be even because it's divisible by 2.\r<BR>\n" );
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document.write( "assuming that n is odd, then there exists a number k such that 2 * k = n + 1.\r<BR>\n" );
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document.write( "n * (n + 1) becomes n * 2 * k which becomes 2 * k * n.\r<BR>\n" );
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document.write( "the number 2 * k * n  has to be even because it's divisible by 2.\r<BR>\n" );
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document.write( "any number that's divisible by 2 is even.\r<BR>\n" );
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document.write( "not sure if this constitutes an acceptable proof by your instructor, but it seems reasonable to me.\r<BR>\n" );
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