document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1070873>Question 1070873</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The second term and the fifth term of a geometric series are 3 and 81 respectively. Find the sum from the fifth term to the tenth term of this series. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #685866 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=685866\">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=81=9%5E2=%283%5E2%29%5E2=3%5E%282%2A2%29=3%5E4 ALIGN=MIDDLE  ALT=\"81=9%5E2=%283%5E2%29%5E2=3%5E%282%2A2%29=3%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Mental math says that the common ratio is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3 ALIGN=MIDDLE  ALT=\"r=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With formulas, id <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B1%5D ALIGN=MIDDLE  ALT=\"b%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the first term,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the common ratio,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Bn%5D=b%5B1%5D%2Ar%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"b%5Bn%5D=b%5B1%5D%2Ar%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B5%5D=b%5B1%5D%2Ar%5E4 ALIGN=MIDDLE  ALT=\"b%5B5%5D=b%5B1%5D%2Ar%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81 ALIGN=MIDDLE  ALT=\"81\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the fifth term,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B2%5D=b%5B1%5D%2Ar ALIGN=MIDDLE  ALT=\"b%5B2%5D=b%5B1%5D%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the second term,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B5%5D%2Fb%5B2%5D=b%5B1%5D%2Ar%5E4%2F%28b%5B1%5D%2Ar%29 ALIGN=MIDDLE  ALT=\"b%5B5%5D%2Fb%5B2%5D=b%5B1%5D%2Ar%5E4%2F%28b%5B1%5D%2Ar%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5B5%5D%2Fb%5B2%5D=r%5E3 ALIGN=MIDDLE  ALT=\"b%5B5%5D%2Fb%5B2%5D=r%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and substituting<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81%2F3=r%5E3 ALIGN=MIDDLE  ALT=\"81%2F3=r%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E4%2F3=r%5E3 ALIGN=MIDDLE  ALT=\"3%5E4%2F3=r%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E3=r%5E3 ALIGN=MIDDLE  ALT=\"3%5E3=r%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3 ALIGN=MIDDLE  ALT=\"r=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The sum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms starting from a term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "in a geometric series with common ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > , is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%28r%5En-1%29%2F%28r-1%29 ALIGN=MIDDLE  ALT=\"b%28r%5En-1%29%2F%28r-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We are skipping the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms of a geometric series with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3 ALIGN=MIDDLE  ALT=\"r=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and adding up the next <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=10-4=6 ALIGN=MIDDLE  ALT=\"n=10-4=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms,<BR>\n" );
document.write( "starting from fifth term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=81 ALIGN=MIDDLE  ALT=\"b=81\"   DISABLED_style= \"max-width:90%; height:auto;\" > and adding up to the tenth term,<BR>\n" );
document.write( "so the sum is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81%283%5E6-1%29%2F%283-1%29=81%28729-1%29%2F2=81%2A728%2F2=81%2A364=highlight%2829484%29 ALIGN=MIDDLE  ALT=\"81%283%5E6-1%29%2F%283-1%29=81%28729-1%29%2F2=81%2A728%2F2=81%2A364=highlight%2829484%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You may think that I need to find the first term of that series,<BR>\n" );
document.write( "then calculate the sum of the first 10 terms,<BR>\n" );
document.write( "next calculate the sum of the first 4 terms,<BR>\n" );
document.write( "and finally subtract one sum from the other.<BR>\n" );
document.write( "That is inefficient.<BR>\n" );
document.write( "Who can say that I am not allowed to think of my own series,<BR>\n" );
document.write( "which happens to have the same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the series in the problem,<BR>\n" );
document.write( "and the same terms, except for missing the first 4? </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );