document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1084552>Question 1084552</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum to m term of the series 1 +11+111...... upto m term is equal to: </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #698706 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=698706\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5Bk%5D+=+%281%2F9%29%2810%5Ek+-+1%29+ ALIGN=MIDDLE  ALT=\"+a%5Bk%5D+=+%281%2F9%29%2810%5Ek+-+1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >    k=1,2,3,…\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5Bn%5D+=+sum+%28++%281%2F9%29%2A%2810%5Ek+-+1%29%2C+k=1%2C+n+%29+ ALIGN=MIDDLE  ALT=\"+S%5Bn%5D+=+sum+%28++%281%2F9%29%2A%2810%5Ek+-+1%29%2C+k=1%2C+n+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5B1%5D++=+1++ ALIGN=MIDDLE  ALT=\"+S%5B1%5D++=+1++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5B2%5D+=+12+ ALIGN=MIDDLE  ALT=\"+S%5B2%5D+=+12+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5B3%5D+=+123+ ALIGN=MIDDLE  ALT=\"+S%5B3%5D+=+123+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+9%2AS%5B1%5D++=+9++ ALIGN=MIDDLE  ALT=\"+9%2AS%5B1%5D++=+9++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+9%2AS%5B2%5D+=+108+ ALIGN=MIDDLE  ALT=\"+9%2AS%5B2%5D+=+108+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+9%2AS%5B3%5D+=+1107++ ALIGN=MIDDLE  ALT=\"+9%2AS%5B3%5D+=+1107++\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Didn't see how 9*S[n] would help, ideally looking for something that is closer to 10^n for each S[n], so we can find a pattern in the amounts to subtract from 10^n to arrive at S[n]:\r<BR>\n" );
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document.write( "—<BR>\n" );
document.write( "Note that  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+81%2AS%5B1%5D+=+81+=+10%5E2+-+10+-+9++ ALIGN=MIDDLE  ALT=\"+81%2AS%5B1%5D+=+81+=+10%5E2+-+10+-+9++\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+81%2AS%5B2%5D+=+972+=+10%5E3+-+10+-+9%2A2+ ALIGN=MIDDLE  ALT=\"+81%2AS%5B2%5D+=+972+=+10%5E3+-+10+-+9%2A2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+81%2AS%5B3%5D+=+9963+=+10%5E4+-+10+-+9%2A3+ ALIGN=MIDDLE  ALT=\"+81%2AS%5B3%5D+=+9963+=+10%5E4+-+10+-+9%2A3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "gives us   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+81%2AS%5Bn%5D+=+10%5E%28n%2B1%29+-10+-+9n+ ALIGN=MIDDLE  ALT=\"+81%2AS%5Bn%5D+=+10%5E%28n%2B1%29+-10+-+9n+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28S%5Bn%5D+=+%281%2F81%29%2810%5E%28n%2B1%29+-+10+-+9n%29%29+ ALIGN=MIDDLE  ALT=\"+highlight%28S%5Bn%5D+=+%281%2F81%29%2810%5E%28n%2B1%29+-+10+-+9n%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "—\r<BR>\n" );
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document.write( "Spot Check:  <BR>\n" );
document.write( "             <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5B1%5D+=+%281%2F81%29%2810%5E%281%2B1%29+-+10+-+9%2A1%29+=+%281%2F81%29%28100-10-9%29+=+81%2F81+=+1+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  (ok)<BR>\n" );
document.write( "             <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+S%5B3%5D+=+%281%2F81%29%2810%5E%283%2B1%29+-+10+-+9%2A3%29+=+%281%2F81%29%2810000-10-9%2A3%29+=+%281%2F81%29%289963%29+=+123+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > (ok) <BR>\n" );
document.write( "—\r<BR>\n" );
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document.write( "I found this by trial and error, maybe another tutor has a more structured approach.<BR>\n" );
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