document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=194091>Question 194091</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the one millionth term in the sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5.....in which each positive integer n occurs n times\r<BR>\n" );
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document.write( "please help me to solve this!!!<BR>\n" );
document.write( " i really don't know how to solve </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #145702 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=145702\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, let's number each term and denote these numbers as \"n\". In other words, 1 is the first term, 2 is the second term, 2 is the third term, 3 is the fourth term, etc... to get\r<BR>\n" );
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document.write( "n:    1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13,  14,  15\r\n" );
document.write( "term: 1,  2,  2,  3,  3,  3,  4,  4,  4,   4,   5,   5,   5,   5,   5\r\n" );
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document.write( "Notice how the LAST number of each grouping lands on the \"n\" values: 1, 3, 6, 10, and 15\r<BR>\n" );
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document.write( "This sequence is the list of triangular numbers which can be found through the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%28m%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28m%28m%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So for example, the last occurrence of the number 3 is when n=6. It turns out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%283%2B1%29%29%2F2=%283%2A4%29%2F2=12%2F2=6 ALIGN=MIDDLE  ALT=\"%283%283%2B1%29%29%2F2=%283%2A4%29%2F2=12%2F2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Also, the last occurrence of the number 4 is when n=10. It turns out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%284%2B1%29%29%2F2=%284%2A5%29%2F2=20%2F2=10 ALIGN=MIDDLE  ALT=\"%284%284%2B1%29%29%2F2=%284%2A5%29%2F2=20%2F2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "etc...\r<BR>\n" );
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document.write( "Now use the same formula to find where the LAST occurrence of 6 will be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%286%2B1%29%29%2F2=%286%2A7%29%2F2=42%2F2=21 ALIGN=MIDDLE  ALT=\"%286%286%2B1%29%29%2F2=%286%2A7%29%2F2=42%2F2=21\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this means that the LAST occurrence of the number 6 will be when n=21 (ie it will be the 21st term). Note: write out the next 6 terms and you'll find that the LAST 6 does fall on n=21\r<BR>\n" );
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document.write( "You can continue this indefinitely. So this means that we can use this formula to find out what the millionth term  of the sequence is.\r<BR>\n" );
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document.write( "It turns out that if you plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1414 ALIGN=MIDDLE  ALT=\"m=1414\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281414%281414%2B1%29%29%2F2=%281414%281415%29%29%2F2=2000810%2F2=1000405 ALIGN=MIDDLE  ALT=\"%281414%281414%2B1%29%29%2F2=%281414%281415%29%29%2F2=2000810%2F2=1000405\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means that the LAST occurrence of the term 1414 is in the location n=1000405. Since 405 is smaller than 1,000,000, this means that the millionth term is also the value 1414.\r<BR>\n" );
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document.write( "Note: to find the value 1414, you can either guess or solve the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%28m%2B1%29%29%2F2=1000000 ALIGN=MIDDLE  ALT=\"%28m%28m%2B1%29%29%2F2=1000000\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find an approximate value of \"m\". From there, round to the nearest whole number.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the millionth term is 1414.\r<BR>\n" );
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