document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099271>Question 1099271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The pattern forming the irrational number 0.543210123450054321000123450000... continues indefinitely. What is the 550th digit in this pattern? Use the leading 5. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #713741 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=713741\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>The other tutor clearly misunderstood the problem.  If the sequence of digits repeated, the number would not be irrational....<br><BR>\n" );
document.write( "The given decimal is<BR>\n" );
document.write( "0.54321 0 12345 00 54321 000 12345 0000 54321 00000 12345 000000 ...<br><BR>\n" );
document.write( "I have added spaces to make it easier to analyze the pattern of the digits.  If we take each \"54321\" as the beginning of a new sub-sequence, then...<BR>\n" );
document.write( "the first sub-sequence has 13 terms - 2 sequences of 5 nonzero digits, and 1+2=3 zeros<BR>\n" );
document.write( "the second sub-sequence has 17 terms - 2 sequences of 5 nonzero digits, and 3+4=7 zeros<BR>\n" );
document.write( "the third sub-sequence has 21 terms - 2 sequences of 5 nonzero digits, and 5+6=11 zeros<br><BR>\n" );
document.write( "Clearly the numbers of digits in the successive sub-sequences form the arithmetic sequence<BR>\n" );
document.write( "13, 17, 21, 25, ...<br><BR>\n" );
document.write( "To solve the problem, we want to know where the end of one of these sub-sequences is shortly before the 550th term of the given sequence.<br><BR>\n" );
document.write( "The n-th term of the arithmetic sequence 13, 17, 21, 25, ... is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2B4%28n-1%29+=+13%2B4n-4+=+4n%2B9 ALIGN=MIDDLE  ALT=\"13%2B4%28n-1%29+=+13%2B4n-4+=+4n%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The sum of the first n terms of an arithmetic sequence is the average of the first term and last term, multiplied by the number of terms,  For this sequence, that is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2813%2B4n%2B9%29%2F2%29%2An+=+2n%5E2%2B11n ALIGN=MIDDLE  ALT=\"%28%2813%2B4n%2B9%29%2F2%29%2An+=+2n%5E2%2B11n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "By one method or another, we find that the largest n for which this is less than 550 is n=14, which gives us a sum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2814%5E2%29%2B11%2814%29+=+546 ALIGN=MIDDLE  ALT=\"2%2814%5E2%29%2B11%2814%29+=+546\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That means the 546th term is the last 0 in one of the sub-sequences.  The 550th term is 4 digits after that, which is the 4th digit in the \"54321\" that starts each sub-sequence.<br><BR>\n" );
document.write( "So the 550th digit in the decimal part of the given irrational number is \"2\".<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );