document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=3461>Question 3461</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The pages of a report are numbered consecutively from 1 to 10. If the sum of the page numbers up to and including page number x of the report is equal to one more than the sum of the page numbers following page number x, then x =\r<BR>\n" );
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document.write( "  (A) 4   (B) 5   (C) 6   (D) 7   (E) 8 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #1533 by </B><A HREF=/tutors/aboutme.mpl?userid=drglass><B>drglass(89)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drglass><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=1533\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To do this problem it is helpful to know that the sum of the numbers from 1 to x is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n+%2B+1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28n%28n+%2B+1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means the sum of the page numbers from 1 to x is:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%28x+%2B+1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28x%28x+%2B+1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "To find the remaining sum, think about it this way <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5+%2B+4+=+1+%2B+2+%2B+3+%2B+4+%2B+5+-+%281+%2B+2+%2B+3%29 ALIGN=MIDDLE  ALT=\"5+%2B+4+=+1+%2B+2+%2B+3+%2B+4+%2B+5+-+%281+%2B+2+%2B+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In other words the sum from (x+1) to 10 is the sum of 1 to 10 minus the sum of 1 to x. This allows us to write a relatively simple expression for the sum from (x+1) to 10<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%2810+%2B+1%29%29%2F2+-+%28x%28x+%2B+1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%2810%2810+%2B+1%29%29%2F2+-+%28x%28x+%2B+1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since the sum from one to x is 1 plus the sum of x+1 to 10, we get the following equation<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%28x+%2B+1%29%29%2F2+=+1+%2B+%2810%2810+%2B+1%29%29%2F2+-+%28x%28x+%2B+1%29%29%2F2+=+1+%2B+55++-+%28x%28x+%2B+1%29%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > or<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%28x+%2B+1%29%29%2F2+=+56++-+%28x%28x+%2B+1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28x%28x+%2B+1%29%29%2F2+=+56++-+%28x%28x+%2B+1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If we add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%28x+%2B+1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28x%28x+%2B+1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides we get<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28x+%2B+1%29+=+56 ALIGN=MIDDLE  ALT=\"x%28x+%2B+1%29+=+56\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The equation becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+x+-+56+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+x+-+56+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which factors nicely to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+7%29%28x+%2B+8%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+-+7%29%28x+%2B+8%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the solution to the equation is x = 7 and x = -8. Well, you cannot have negative pages, so the answer to the question must be 7. Let's check it.<BR>\n" );
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document.write( "the sum from 1 to 7 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%287%2B1%29%29%2F2+=+28 ALIGN=MIDDLE  ALT=\"%287%287%2B1%29%29%2F2+=+28\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8+%2B+9+%2B+10+=+27 ALIGN=MIDDLE  ALT=\"8+%2B+9+%2B+10+=+27\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is one more than the sum from 1 to 7. So the answer is indeed 7. </SPAN>\n" );
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