document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1143590>Question 1143590</A>: <I><SPAN STYLE=\"word-break: break-all;\"> given a and b, suppose that three numbers are inserted between them so that the five numbers form a geometric sequence. if the product of the three numbers between a and b is 27, show that ab is equal to 9. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #764434 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52798)</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=764434\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
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document.write( "Then \"the product of the tree numbers between a and b\" is the cube of the very middle term, which, hence, is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C27%29 ALIGN=MIDDLE  ALT=\"root%283%2C27%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 3.\r\n" );
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document.write( "Further, the three numbers  \"a\", 3  and \"b\"  form a geometric progression.\r\n" );
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document.write( "For every geometric progression, the middle of any three consecutive terms is the geometric mean of the two extreme terms.\r\n" );
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document.write( "So,  then  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28ab%29 ALIGN=MIDDLE  ALT=\"sqrt%28ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 3,  which implies  ab = 9.\r\n" );
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document.write( "Q.E.D.\r<BR>\n" );
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document.write( "There is a bunch of lessons on geometric progressions 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=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=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/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( "Especially, see the lesson (*) from the list.\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( "Save the link to this textbook together with its description\r<BR>\n" );
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document.write( "Free of charge online textbook in ALGEBRA-II<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r<BR>\n" );
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document.write( "into your archive and use when it is needed.\r<BR>\n" );
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