document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209783>Question 1209783</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate<BR>\n" );
document.write( "1 + \frac{i}{3} - \frac{1}{9} - \frac{i}{27} + \frac{1}{81},<BR>\n" );
document.write( "where $i$ is the imaginary unit.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850414 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=850414\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bi%2F3-1%2F9-i%2F27%2B1%2F81 ALIGN=MIDDLE  ALT=\"1%2Bi%2F3-1%2F9-i%2F27%2B1%2F81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Combine like terms -- i.e., combine the real terms and combine the imaginary terms.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-1%2F9%2B1%2F81%29%2Bi%281%2F3-1%2F27%29 ALIGN=MIDDLE  ALT=\"%281-1%2F9%2B1%2F81%29%2Bi%281%2F3-1%2F27%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2881%2F81-9%2F81%2B1%2F81%29%2Bi%289%2F27-1%2F27%29 ALIGN=MIDDLE  ALT=\"%2881%2F81-9%2F81%2B1%2F81%29%2Bi%289%2F27-1%2F27%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2873%2F81%29%2Bi%288%2F27%29 ALIGN=MIDDLE  ALT=\"%2873%2F81%29%2Bi%288%2F27%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: (73/81)+(8/27)i<br><BR>\n" );
document.write( "It is possible that the sequence was supposed to be an infinite sequence instead of a finite one.  In that case....<br><BR>\n" );
document.write( "Sum = (first term)/(1-common difference)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281-i%2F3%29=1%2F%28%283-i%29%2F3%29=3%2F%283-i%29=%283%283%2Bi%29%29%2F10=%289%2B3i%29%2F10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: (9+3i)/10<br><BR>\n" );
document.write( "Alternatively, we can find the infinite sums of the real and imaginary parts separately.<br><BR>\n" );
document.write( "Real parts: first term 1, common ratio (-1/9)<br><BR>\n" );
document.write( "Sum = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%281-%28-1%2F9%29%29=1%2F%281%2B1%2F9%29=1%2F%2810%2F9%29=9%2F10 ALIGN=MIDDLE  ALT=\"1%2F%281-%28-1%2F9%29%29=1%2F%281%2B1%2F9%29=1%2F%2810%2F9%29=9%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Imaginary parts: first term i/3, common ratio (-1/9)<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28i%2F3%29%2F%281-%28-1%2F9%29%29=%28i%2F3%29%2F%281%2B1%2F9%29=%28i%2F3%29%2F%2810%2F9%29=%28i%2F3%29%289%2F10%29=3i%2F10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: (9/10)+(3/10)i<br><BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );