document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1091273>Question 1091273</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find the two possible fourth values of a geometric sequence.  There are two geometric sequences with 3rd term 5 and 5th term 80.  What are the possible 4th terms? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #705662 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52790)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=705662\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D ALIGN=MIDDLE  ALT=\"a%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Er%5E2 ALIGN=MIDDLE  ALT=\"a%5Er%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 5     for the 3-rd term.\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B5%5D ALIGN=MIDDLE  ALT=\"a%5B5%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Er%5E4 ALIGN=MIDDLE  ALT=\"a%5Er%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 80    for the 5-th term.\r\n" );
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document.write( "Take the ratio\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B5%5D%2Fa%5B3%5D ALIGN=MIDDLE  ALT=\"a%5B5%5D%2Fa%5B3%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=80%2F5 ALIGN=MIDDLE  ALT=\"80%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 16  = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Ar%5E4%29%2F%28a%2Ar%5E2%29 ALIGN=MIDDLE  ALT=\"%28a%2Ar%5E4%29%2F%28a%2Ar%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2 ALIGN=MIDDLE  ALT=\"r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "Thus you have an equation for \"r\":  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2 ALIGN=MIDDLE  ALT=\"r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 16,  which has TWO solutions:  r = 4   and/or  r = -4.\r\n" );
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document.write( "So, the two possibilities for the 4-th term are\r\n" );
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document.write( "1)  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D ALIGN=MIDDLE  ALT=\"a%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D%2Ar ALIGN=MIDDLE  ALT=\"a%5B3%5D%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 5*4    = 20,   and\r\n" );
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document.write( "2)  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D ALIGN=MIDDLE  ALT=\"a%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D%2Ar ALIGN=MIDDLE  ALT=\"a%5B3%5D%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 5*(-4) = -20.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "On geometric progressions,  see the lessons in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Geometric-progressions.lesson>Geometric progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-geometric-progressions.lesson>The proofs of the formulas for geometric progressions</A> \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-geometric-progressions.lesson>Problems on geometric progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-geometric-progressions.lesson>Word problems on geometric progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-geometric-progressions.lesson>One characteristic property of geometric progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-geometric-progressions.lesson>Solved problems on geometric progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Fresh-sweet-and-crispy-problem-on-arithmetic-and-geometric-progressions.lesson>Fresh, sweet and crispy problem on arithmetic and geometric progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-geometric-progressions.lesson>Mathematical induction and geometric progressions</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-for-sequences-other-than-arithmetic-or-geometric.lesson>Mathematical induction for sequences other than arithmetic or geometric</A>\r<BR>\n" );
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document.write( "Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lessons are the part of this online textbook under the topic <U>\"Geometric progressions\"</U>.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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