document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252455>Question 252455</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate: What is the sum of all the one’s digits in the following finite sequence?\r<BR>\n" );
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document.write( "3^0 ,3^1,3^2 ,3^3 ,...,3^2006 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184374 by </B><A HREF=/tutors/aboutme.mpl?userid=drk><B>drk(1908)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drk><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=184374\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There is a pattern for the 3's\r<BR>\n" );
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document.write( "3^1 = 3<BR>\n" );
document.write( "3^2 = 9<BR>\n" );
document.write( "3^3 = 27<BR>\n" );
document.write( "3^4 = 81.<BR>\n" );
document.write( "3^5 = 243<BR>\n" );
document.write( "Notice the pattern {3,9,7,1} repeats after every fourth group. The sum of 3,9,7,and 1 is 20. 2006 / 4 will get you the number of groups. That quotient is 501 remainder 2.\r<BR>\n" );
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document.write( "20*501 = 10020. This is the sum of the first 501 groups. Remainder 2 means add 3+9, or 3^2005 + 3^2006 = 3^1 + 3^2 in terms of units digits.\r<BR>\n" );
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document.write( "So, 20*501 + 12 = 10032. Don't forget to add back in 3^0 = 1. <BR>\n" );
document.write( "The units digit is 2+1 = 3. </SPAN>\n" );
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