document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=637274>Question 637274</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of the first ten terms of the the series 1+4+16+64+256 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #401566 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=401566\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The terms are in a geometric sequence, or geometric progression.<BR>\n" );
document.write( "Each term is the previous one multiplied times <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( "You may be studying geometric sequences, also called geometric progressions.<BR>\n" );
document.write( "The symbols and names used vary a little from place to place, from book to book, and from teacher to teacher, so the symbols, and formulas may look different, but the idea is the same. I will use a popular choice for terms, symbols, and formulas. Keep in mind that your book/teacher may use slightly different ones.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE GENERAL THEORY (what the book may say):<BR>\n" );
document.write( "In a geometric sequence, or geometric progression, the term number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-1 ALIGN=MIDDLE  ALT=\"n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the previous one) are related by the recurrence formulas<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%5Bn-1%5D%2Ar ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5Bn-1%5D%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=a%5Bn%5D%2F%28a%5Bn-1%5D%29 ALIGN=MIDDLE  ALT=\"r=a%5Bn%5D%2F%28a%5Bn-1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "Where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the tern number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn-1%5D ALIGN=MIDDLE  ALT=\"a%5Bn-1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the previous term, 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( "The term number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be calculated from the first term and the common ratio, using the explicit formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%5B1%5D%2Ar%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B1%5D%2Ar%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sum of the first <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, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=a%5B1%5D%2Ba%5B2%5D ALIGN=MIDDLE  ALT=\"S%5Bn%5D=a%5B1%5D%2Ba%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ ... +<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn-1%5D+%2B+a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn-1%5D+%2B+a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=a%5B1%5D%28r%5En-1%29%2F%28r-1%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=a%5B1%5D%28r%5En-1%29%2F%28r-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The formula can be deduced as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=a%5B1%5D%2Br%2Aa%5B1%5D%2Br%5E2%2Aa%5B1%5D ALIGN=MIDDLE  ALT=\"S%5Bn%5D=a%5B1%5D%2Br%2Aa%5B1%5D%2Br%5E2%2Aa%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ ... +<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E%28n-2%29%2Aa%5B1%5D%2Br%5E%28n-1%29%2Aa%5B1%5D ALIGN=MIDDLE  ALT=\"r%5E%28n-2%29%2Aa%5B1%5D%2Br%5E%28n-1%29%2Aa%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2AS%5Bn%5D=r%2Aa%5B1%5D%2Br%5E2%2Aa%5B1%5D ALIGN=MIDDLE  ALT=\"r%2AS%5Bn%5D=r%2Aa%5B1%5D%2Br%5E2%2Aa%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >+ ... +<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5En%2Aa%5B1%5D%2Br%5E%28n-1%29%2Aa%5B1%5D%2Br%5En%2Aa%5B1%5D ALIGN=MIDDLE  ALT=\"r%5En%2Aa%5B1%5D%2Br%5E%28n-1%29%2Aa%5B1%5D%2Br%5En%2Aa%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting one equation from the other, a lot of terms cancel, and you have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2AS%5Bn%5D-S%5Bn%5D=r%5En%2Aa%5B1%5D-a%5B1%5D ALIGN=MIDDLE  ALT=\"r%2AS%5Bn%5D-S%5Bn%5D=r%5En%2Aa%5B1%5D-a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r-1%29%2AS%5Bn%5D=%28r%5En-1%29%2Aa%5B1%5D ALIGN=MIDDLE  ALT=\"%28r-1%29%2AS%5Bn%5D=%28r%5En-1%29%2Aa%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, dividing both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r-1 ALIGN=MIDDLE  ALT=\"r-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r-1%29%2AS%5Bn%5D=%28-1%29r%5En%2Aa%5B1%5D ALIGN=MIDDLE  ALT=\"%28r-1%29%2AS%5Bn%5D=%28-1%29r%5En%2Aa%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=a%5B1%5D%28r%5En-1%29%2F%28r-1%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=a%5B1%5D%28r%5En-1%29%2F%28r-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Someone discovered that formula. I just remember the trick and re-discover the formula every time, because I do not need to use it often, and do not remember it.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE SOLUTION TO THE PROBLEM:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=1 ALIGN=MIDDLE  ALT=\"a%5B1%5D=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=4 ALIGN=MIDDLE  ALT=\"r=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=10 ALIGN=MIDDLE  ALT=\"n=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Applying the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=a%5B1%5D%28r%5En-1%29%2F%28r-1%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=a%5B1%5D%28r%5En-1%29%2F%28r-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=1%2A%284%5E10-1%29%2F%284-1%29=%284%5E10-1%29%2F3=%281048576-1%29%2F3=1048575%2F3=highlight%28349525%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );