document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1076171>Question 1076171</A>: <I><SPAN STYLE=\"word-break: break-all;\"> R0: 3,6,9,12,15,18,21,...<BR>\n" );
document.write( "R1: 1,4,7,10,13,16,19,...<BR>\n" );
document.write( "R2: 2,5,8,11,14,17,20,...<BR>\n" );
document.write( "a) Write down an expression for the value of the general term in each of the three prgressions. Show that the integer 1706836 lies in R1. I got R0:3n, R1:3n-2 and R2:3n-1. Then i did 3n-2=1706836 and showed that n=568946, so it lies in R1 and is the 568946th term in the series.<BR>\n" );
document.write( "b) Use the Binomial Theorem to show that if x is a term in R1 or R2 then x^6 is in R1. This is the part i don't know how to work out.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #690841 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=690841\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) I agree with your expressions for the the general term in each of the three sequences.<BR>\n" );
document.write( "To show if any number belongs to one of them,<BR>\n" );
document.write( "I would just add digits, using module 9 clock arithmetic. The number belongs in the sequence that contains the final sum.<BR>\n" );
document.write( "If the ending sum is 0, 3, or 6, the number is divisible by 3, and belongs in R0.<BR>\n" );
document.write( "If it is 1, 4, or 7, it belongs in R1; if 2,5, or 8, it belongs in R2.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1706836 ALIGN=MIDDLE  ALT=\"1706836\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the final <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36 ALIGN=MIDDLE  ALT=\"36\"   DISABLED_style= \"max-width:90%; height:auto;\" > part adds up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B69 ALIGN=MIDDLE  ALT=\"3%2B69\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that 9 is dropped.<BR>\n" );
document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B8=9 ALIGN=MIDDLE  ALT=\"1%2B8=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and that 9 is dropped.<BR>\n" );
document.write( "Then we are left with  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B0%2B6=13 ALIGN=MIDDLE  ALT=\"7%2B0%2B6=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B3=4 ALIGN=MIDDLE  ALT=\"1%2B3=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we conclude that<BR>\n" );
document.write( "when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1706836 ALIGN=MIDDLE  ALT=\"1706836\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divided by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the remainder is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1706836=9K%2B4=3%2A3K%2B6-2=3%283K%2B2%29-2 ALIGN=MIDDLE  ALT=\"1706836=9K%2B4=3%2A3K%2B6-2=3%283K%2B2%29-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K ALIGN=MIDDLE  ALT=\"K\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that shows that it belongs in R1.<BR>\n" );
document.write( " <BR>\n" );
document.write( "HOW AND WHY CLOCK ARITHMETIC MOD 9 WORKS:<BR>\n" );
document.write( "The sum of the digits of a number divisible by 9 is 9 or a multiple of 9.<BR>\n" );
document.write( "As you are adding,<BR>\n" );
document.write( "you drop partial sums that are 9, and keep adding the rest of the digits.<BR>\n" );
document.write( "Similarly, for any partial two-digit sum,<BR>\n" );
document.write( "you replace the sum of the digits for the two-digit partial sum.<BR>\n" );
document.write( "At the end, you should have a one-digit final sum from 0 to 8.<BR>\n" );
document.write( " <BR>\n" );
document.write( "b)The numbers in R1 and R2 are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > away from a multiple of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We could write them all as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p+%2B-+1 ALIGN=MIDDLE  ALT=\"3p+%2B-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some non-negative integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( ".<BR>\n" );
document.write( "(If you - or your teacher - do not like that expression, you could use<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3p+%2B+1 ALIGN=MIDDLE  ALT=\"3p+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for terms of R1, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=n-1 ALIGN=MIDDLE  ALT=\"p=n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n+-+1 ALIGN=MIDDLE  ALT=\"3n+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for terms of R2,<BR>\n" );
document.write( "and write twice as much for your proof).<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3p+%2B-+1 ALIGN=MIDDLE  ALT=\"x=3p+%2B-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E6=%283p+%2B-+1%29%5E6=%283p%29%5E6+%2B-+6%2A%281%29%283p%29%5E5 ALIGN=MIDDLE  ALT=\"x%5E6=%283p+%2B-+1%29%5E6=%283p%29%5E6+%2B-+6%2A%281%29%283p%29%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%283p%29%5E4%281%29%5E2+%2B-+20%283p%29%5E3%281%29%5E3 ALIGN=MIDDLE  ALT=\"15%283p%29%5E4%281%29%5E2+%2B-+20%283p%29%5E3%281%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%283p%29%5E2%281%29%5E4+%2B-+6%283p%29%281%29%5E5%2B1%5E6 ALIGN=MIDDLE  ALT=\"15%283p%29%5E2%281%29%5E4+%2B-+6%283p%29%281%29%5E5%2B1%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283p%29%5E6+%2B-+6%283p%29%5E5 ALIGN=MIDDLE  ALT=\"%283p%29%5E6+%2B-+6%283p%29%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%283p%29%5E4+%2B-+20%283p%29%5E3 ALIGN=MIDDLE  ALT=\"15%283p%29%5E4+%2B-+20%283p%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%283p%29%5E2+%2B-+6%283p%29+ ALIGN=MIDDLE  ALT=\"15%283p%29%5E2+%2B-+6%283p%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B+1%22 ALIGN=MIDDLE  ALT=\"%22%2B+1%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It should be obvious that all terms of that sum are multiples of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "except that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22+%2B+1%22 ALIGN=MIDDLE  ALT=\"%22+%2B+1%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the end,<BR>\n" );
document.write( "and that shows that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E6 ALIGN=MIDDLE  ALT=\"x%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a term in R1.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If your teacher insisted on seeing the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > that gives <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E6 ALIGN=MIDDLE  ALT=\"x%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > the right to belong to R1,<BR>\n" );
document.write( "you would have to write more, starting with getting out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > as a common factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E6+=+3 ALIGN=MIDDLE  ALT=\"x%5E6+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5B%22 ALIGN=MIDDLE  ALT=\"%22%5B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2A%283p%29%5E5+%2B-+2%283p%29%5E5 ALIGN=MIDDLE  ALT=\"p%2A%283p%29%5E5+%2B-+2%283p%29%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%283p%29%5E4+%2B-+20p%283p%29%5E2 ALIGN=MIDDLE  ALT=\"5%283p%29%5E4+%2B-+20p%283p%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B%22 ALIGN=MIDDLE  ALT=\"%22%2B%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%283p%29%5E2%281%29%5E4+%2B-+2%283p%29 ALIGN=MIDDLE  ALT=\"5%283p%29%5E2%281%29%5E4+%2B-+2%283p%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%5D%22 ALIGN=MIDDLE  ALT=\"%22%5D%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2B+1%22 ALIGN=MIDDLE  ALT=\"%22%2B+1%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );