document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=371140>Question 371140</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sequence goes 1,2,2,3,3,3,... <BR>\n" );
document.write( "Find the 66th term. Please show all the steps and explain why it is like that<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #264415 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=264415\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The number 1 is the first term.\r<BR>\n" );
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document.write( "The last number of 2 is term number 2+1=3 \r<BR>\n" );
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document.write( "The last number of 3 is term number 3+2+1=6 \r<BR>\n" );
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document.write( "The last number of 4 is term number 4+3+2+1=10 (expand it out to get 1,2,2,3,3,3,4,4,4,4 and you'll see that the last 4 is the 10th term)\r<BR>\n" );
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document.write( "The last number of 5 is term number 5+4+3+2+1=15 (expand it out to get 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5 and you'll see that the last 5 is the 15th term)\r<BR>\n" );
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document.write( "This pattern continues....\r<BR>\n" );
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document.write( "If you keep going, you'll get the sequence: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55\r<BR>\n" );
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document.write( "(ie the 21 refers to the position of the last 6)\r<BR>\n" );
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document.write( "Now because 1+2+3+4+5+6+7+8+9+10+11 = 66, this means that the last 11 will be in the 66th position.\r<BR>\n" );
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document.write( "So the 66th term is 11.\r<BR>\n" );
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document.write( "To verify, list out the terms to get\r<BR>\n" );
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document.write( "1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11\r<BR>\n" );
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document.write( "and if you count each term, you'll find that the last term in that list is indeed the 66th term (which is 11)\r<BR>\n" );
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document.write( "Hopefully, this makes sense. If not, please let me know. </SPAN>\n" );
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