document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=389515>Question 389515</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let a_n = 11...1 with 3^n digits. Prove that a_n is divisible by 3a_(n-1). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #276066 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=276066\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a_1 = 111 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=111%2A10%5E0+=+111%2A%28%2810%5E3+-+1%29%2F%2810%5E3+-+1%29%29 ALIGN=MIDDLE  ALT=\"111%2A10%5E0+=+111%2A%28%2810%5E3+-+1%29%2F%2810%5E3+-+1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a_2 = 111 111 111 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=111%2A%28%28%2810%5E3%29%5E3+-+1%29%2F%2810%5E3+-+1%29%29+=+111%2A%28%2810%5E%283%5E2%29+-+1%29%2F%2810%5E3+-+1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a_3 = 111 111 111...111  (27 1's) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=111%2A%28%28%2810%5E3%29%5E%283%5E2%29+-+1%29%2F%2810%5E3+-+1%29%29+=+111%2A%28%2810%5E%283%5E3%29+-+1%29%2F%2810%5E3+-+1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "........<BR>\n" );
document.write( "a_n = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=111%2A%28%28%2810%5E3%29%5E%283%5E%28n-1%29%29+-+1%29%2F%2810%5E3+-+1%29%29+=+111%2A%28%2810%5E%283%5En%29+-+1%29%2F%2810%5E3+-+1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > by induction.\r<BR>\n" );
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document.write( "Then<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a_n%2F%283a_%28n-1%29%29+%0D%0A=+111%2A%28%2810%5E%283%5En%29+-+1%29%2F999%29%2F%28+3%2A111%2A%28%2810%5E%283%5E%28n-1%29%29+-+1%29%2F999%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, substituting 999 for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%5E3+-+1 ALIGN=MIDDLE  ALT=\"10%5E3+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a_n%2F%283a_%28n-1%29%29++=+%2810%5E%283%5En%29+-+1%29%2F%283%2A%2810%5E%283%5E%28n-1%29%29+-+1%29%29+=+%2810%5E%282%2A3%5E%28n-1%29%29+%2B+10%5E%283%5E%28n-1%29%29+%2B+1%29%2F3 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%5E%282%2A3%5E%28n-1%29%29+%2B+10%5E%283%5E%28n-1%29%29+%2B+1%29%2F3 ALIGN=MIDDLE  ALT=\"%2810%5E%282%2A3%5E%28n-1%29%29+%2B+10%5E%283%5E%28n-1%29%29+%2B+1%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is definitely a positive integer, because the numerator is a positive integer with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A3%5E%28n-1%29%2B1 ALIGN=MIDDLE  ALT=\"2%2A3%5E%28n-1%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > digits, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A3%5E%28n-1%29+-+2 ALIGN=MIDDLE  ALT=\"2%2A3%5E%28n-1%29+-+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > of which are 0's, and 3 are 1's.  Hence the sum of the digits of the numerator is 3, which makes the number itself divisible by 3.<BR>\n" );
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