document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=288282>Question 288282</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The  value   of  - 2 + 3 - 4 + 5 - 6 +. . . - 100   is\r<BR>\n" );
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document.write( "(a) -150 (b) =100 (c) -50 (d) -49 (e) 0\r<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 #209045 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=209045\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hint: Rewrite the series - 2 + 3 - 4 + 5 - 6 +. . . - 100 into (3+5+7+9+...97+99)+(-2-4-6-8-...-98-100). From there, break up 3 into 2*1+1, break up 5 into 2*2+1, break up 7 into 2*3+1, etc to get 2*1+1+2*2+1+2*3+1+...2*48+1+2*49+1. Also, rewrite 2, 4, 6, etc into 2*1, 2*2, 2*3, etc\r<BR>\n" );
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document.write( "So...\r<BR>\n" );
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document.write( "- 2 + 3 - 4 + 5 - 6 +. . . - 100 = (3+5+7+9+...97+99)+(-2-4-6-8-...-98-100) = (2*1+1+2*2+1+2*3+1+...2*48+1+2*49+1) + (-2*1-2*2-2*3-...-2*49-2*50)\r<BR>\n" );
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document.write( "Or simply \r<BR>\n" );
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document.write( "- 2 + 3 - 4 + 5 - 6 +. . . - 100 = (2*1+1+2*2+1+2*3+1+...2*48+1+2*49+1) + (-2*1-2*2-2*3-...-2*49-2*50)\r<BR>\n" );
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document.write( "Now group all of the terms with a factor of 2 in (2*1+1+2*2+1+2*3+1+...2*48+1+2*49+1) to get ((2*1+2*2+2*3+...2*48+2*49)+1+1+1+...1+1)\r<BR>\n" );
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document.write( "Now we then get \r<BR>\n" );
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document.write( "- 2 + 3 - 4 + 5 - 6 +. . . - 100 = ((2*1+2*2+2*3+...2*48+2*49)+1+1+1+...1+1) + (-2*1-2*2-2*3-...-2*49-2*50)\r<BR>\n" );
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document.write( "Factor out the '2's:\r<BR>\n" );
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document.write( "- 2 + 3 - 4 + 5 - 6 +. . . - 100 = (2(1+2+3+...48+49)+1+1+1+...1+1) + -2(1+2+3+...+49+50)\r<BR>\n" );
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document.write( "What we now have is a series 1+2+3+....+n in which we can use the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28n%28n%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. I'll let you do that. </SPAN>\n" );
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