document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1148539>Question 1148539</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the first m positive odd integers is 212 more than the sum of the first n positive even integers. What is the sum of all possible values of n? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #769891 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=769891\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The sum of the first m positive odd integers is m^2.<br><BR>\n" );
document.write( "The sum of the first n positive even integers is n^2+n.<br><BR>\n" );
document.write( "So the requirement is that m^2 = n^2+n+212; i.e., the square root of (n^2+n+212) must be an integer.<br><BR>\n" );
document.write( "I don't know an algebraic way to find solutions.<br><BR>\n" );
document.write( "An excel spreadsheet for calculating the square root of n^2+n+212 shows three solutions:<br><BR>\n" );
document.write( "n = 16; m = 22<BR>\n" );
document.write( "n = 28; m = 32<BR>\n" );
document.write( "n = 211; m = 212<br><BR>\n" );
document.write( "So the answer to the given question is 16+28+211 = 255.<br><BR>\n" );
document.write( "I will be interested to see if any of the other tutors have an algebraic method for finding those solutions.... </SPAN>\n" );
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