document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209803>Question 1209803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let<BR>\n" );
document.write( "a + ar + ar^2 + ar^3 + \dotsb<BR>\n" );
document.write( "be an infinite geometric series. The sum of the series is 9. The sum of the cubes of all the terms is 36. Find the common ratio. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850593 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850593\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Sum of the Infinite Geometric Series**\r<BR>\n" );
document.write( "\n" );
document.write( "* The sum of an infinite geometric series is given by: S = a / (1 - r)<BR>\n" );
document.write( "* We are given that the sum is 9: a / (1 - r) = 9\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Sum of the Cubes of the Terms**\r<BR>\n" );
document.write( "\n" );
document.write( "* The terms of the series are: a, ar, ar², ar³, ...<BR>\n" );
document.write( "* Cubing each term: a³, a³r³, a³r⁶, a³r⁹, ...<BR>\n" );
document.write( "* This is also a geometric series with:<BR>\n" );
document.write( "    * First term: a³<BR>\n" );
document.write( "    * Common ratio: r³<BR>\n" );
document.write( "* The sum of this series is: a³ / (1 - r³)<BR>\n" );
document.write( "* We are given that this sum is 36: a³ / (1 - r³) = 36\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Solve the System of Equations**\r<BR>\n" );
document.write( "\n" );
document.write( "* From a / (1 - r) = 9, we get: a = 9(1 - r)<BR>\n" );
document.write( "* Substitute this into a³ / (1 - r³) = 36:<BR>\n" );
document.write( "    * [9(1 - r)]³ / (1 - r³) = 36<BR>\n" );
document.write( "    * 729(1 - r)³ / (1 - r³) = 36<BR>\n" );
document.write( "    * (1 - r)³ / (1 - r³) = 36 / 729 = 4 / 81\r<BR>\n" );
document.write( "\n" );
document.write( "* Factor (1 - r³):<BR>\n" );
document.write( "    * (1 - r³) = (1 - r)(1 + r + r²)<BR>\n" );
document.write( "* Substitute this into the equation:<BR>\n" );
document.write( "    * (1 - r)³ / [(1 - r)(1 + r + r²)] = 4 / 81<BR>\n" );
document.write( "    * (1 - r)² / (1 + r + r²) = 4 / 81\r<BR>\n" );
document.write( "\n" );
document.write( "* Take the square root of both sides:<BR>\n" );
document.write( "    * (1 - r) / √(1 + r + r²) = ± 2 / 9\r<BR>\n" );
document.write( "\n" );
document.write( "* We'll assume r is positive, so we'll take the positive root.\r<BR>\n" );
document.write( "\n" );
document.write( "* (1 - r) / √(1 + r + r²) = 2 / 9<BR>\n" );
document.write( "* 9(1 - r) = 2√(1 + r + r²)<BR>\n" );
document.write( "* 81(1 - r)² = 4(1 + r + r²)<BR>\n" );
document.write( "* 81(1 - 2r + r²) = 4 + 4r + 4r²<BR>\n" );
document.write( "* 81 - 162r + 81r² = 4 + 4r + 4r²<BR>\n" );
document.write( "* 77r² - 166r + 77 = 0\r<BR>\n" );
document.write( "\n" );
document.write( "* Solve the quadratic equation:<BR>\n" );
document.write( "   * r = [166 ± √(166² - 4 * 77 * 77)] / (2 * 77)<BR>\n" );
document.write( "   * r = [166 ± √(27556 - 23716)] / 154<BR>\n" );
document.write( "   * r = [166 ± √3840] / 154<BR>\n" );
document.write( "   * r = [166 ± 62.0967] / 154\r<BR>\n" );
document.write( "\n" );
document.write( "* We have two possible values for r:<BR>\n" );
document.write( "    * r1 = (166 + 62.0967) / 154 ≈ 1.481<BR>\n" );
document.write( "    * r2 = (166 - 62.0967) / 154 ≈ 0.675\r<BR>\n" );
document.write( "\n" );
document.write( "* Since the series is convergent, we must have |r| < 1. Therefore, r = 0.675.\r<BR>\n" );
document.write( "\n" );
document.write( "**4. Check the Solution**\r<BR>\n" );
document.write( "\n" );
document.write( "* r = 0.675 = 27/40<BR>\n" );
document.write( "* a = 9(1 - r) = 9(1 - 27/40) = 9(13/40) = 117/40<BR>\n" );
document.write( "* a³ / (1 - r³) = (117/40)³ / (1 - (27/40)³) ≈ 36\r<BR>\n" );
document.write( "\n" );
document.write( "**Final Answer:** The common ratio is 27/40 or 0.675.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );