document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=285791>Question 285791</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the integers from 1 to 100 inclusive. What is the difference between the sum of all the even integers and the sum of all the odd integers?\r<BR>\n" );
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document.write( "A. 1 B. 2 C. 0 D. 4 • E. 50 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #207209 by </B><A HREF=/tutors/aboutme.mpl?userid=nabla><B>nabla(475)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nabla><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=207209\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's denote our odds as:<BR>\n" );
document.write( "SUM(k=0->49)[2k+1]<BR>\n" );
document.write( "and our evens as:<BR>\n" );
document.write( "SUM(k=0->50)[2k]\r<BR>\n" );
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document.write( "So, we want to evaluate:<BR>\n" );
document.write( "SUM(k=0->50)[2k]-SUM(k=0->49)[2k+1]\r<BR>\n" );
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document.write( "this is equivalent to:<BR>\n" );
document.write( "2*SUM(k=0->50)[k]-2SUM(k=0->49)[k]-50<BR>\n" );
document.write( "which is moreover equivalent to:<BR>\n" );
document.write( "2*50+2*SUM(k=0->49)[k]-2SUM(k=0->49)[k]-50\r<BR>\n" );
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document.write( "Our summation notation cancels, and we are left with 100-50=50. So we shall pick E as the solution.\r<BR>\n" );
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document.write( "If these properties of series are unfamiliar, note first that SUM(k=0->49)[1] is 1 added up 50 times (=50). Moreover, When we easily remove first or last members of the series by simply adding the term, hence 2*SUM(k=0->50)[k]=2*50+SUM(k=0->49)[k].\r<BR>\n" );
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document.write( "Hope this helps. </SPAN>\n" );
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