document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=853347>Question 853347</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the rule for the sum of the first 'n' positive integers.\r<BR>\n" );
document.write( "\n" );
document.write( "I.e. 1+2+3+4+5... \r<BR>\n" );
document.write( "\n" );
document.write( "Could you please give the formula and explain the solution?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #514068 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=514068\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the sum of the first n positive integers is given by the formula<BR>\n" );
document.write( "n(n+1) / 2<BR>\n" );
document.write( "note that if n = 1, then<BR>\n" );
document.write( "1(1+1) / 2 = 2/2 = 1<BR>\n" );
document.write( "now assume that the formula<BR>\n" );
document.write( "n(n+1) / 2 is true for n > 1<BR>\n" );
document.write( "we need to show that<BR>\n" );
document.write( "sum of i where i =1, n+1 is (n+1)(n+2) / 2<BR>\n" );
document.write( "pull n+1 out of the sum and we get<BR>\n" );
document.write( "sum of i where i =1, n+1 is n+1 + sum of i where i =1,n is (n+1)+ n(n+1)/2<BR>\n" );
document.write( "factor out (n+1) and simplify<BR>\n" );
document.write( "(n+1) + n(n+1) / 2 = (n+1)(1+ n/2) = (n+1)(2+n)/2 = (n+1)(n+2) / 2<BR>\n" );
document.write( "which proves the result<BR>\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );