document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=265111>Question 265111</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Why does a perfect square result from adding two successive rows of even and odd integers? Example: 1+2 plus 1+2+3 = 9 <BR>\n" );
document.write( "                       1+2+3 plus 1+2+3+4 = 16 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #194987 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=194987\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let n = last number in first row\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If we generalize the problem, we get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1+2+3+...+n<BR>\n" );
document.write( "1+2+3+...+n+(n+1)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add the two rows to get\r<BR>\n" );
document.write( "\n" );
document.write( "(1+2+3+...+n)+(1+2+3+...+n+(n+1))\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now simplify:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(1+1)+(2+2)+(3+3)+...+(n+n)+n+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2(1)+2(2)+2(3)+...+2(n)+n+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2(1+2+3+...+n)+n+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now use the formula 1+2+3+...+n = n(n+1)/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2(n(n+1)/2)+n+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "n(n+1)+n+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "n^2+n+n+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "n^2+2n+1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(n+1)^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So this shows that the sum of 1+2+3+...+n and 1+2+3+...+n+(n+1) is a perfect square and it shows us that the perfect square is (n+1)^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In the case that n = 4, we get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(1+2+3+4)+(1+2+3+4+5) = 10+15 = 25\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Take note that 25 = 5^2 = (4+1)^2 which confirms our answer.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );