document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083522>Question 1083522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the missing term or terms in each geometric sequence.<BR>\n" );
document.write( "...,4,_,_,108, ... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697526 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=697526\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" size=\"4\"><BR>\n" );
document.write( "Answer for the first blank: <font color=red>12</font><BR>\n" );
document.write( "Answer for the second blank: <font color=red>36</font><BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "The first term of this sequence is 4.\r<BR>\n" );
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document.write( "The second term is unknown. <BR>\n" );
document.write( "While we don't know it, we do know that it's related to the first term. <BR>\n" );
document.write( "Specifically it is equal to 4*r where r is the common ratio of this geometric sequence.<BR>\n" );
document.write( "In other words,<BR>\n" );
document.write( "second term = (common ratio)*(first term)<BR>\n" );
document.write( "second term = (r)*(4)<BR>\n" );
document.write( "second term = 4*r<BR>\n" );
document.write( "Notice how the second term is built on the first term. In general, any given term is dependent on the previous term (as you'll see below).\r<BR>\n" );
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document.write( "Similarly,<BR>\n" );
document.write( "third term = (common ratio)*(second term)<BR>\n" );
document.write( "third term = (r)*(4*r)<BR>\n" );
document.write( "third term = 4r^2\r<BR>\n" );
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document.write( "and,<BR>\n" );
document.write( "fourth term = (common ratio)*(third term)<BR>\n" );
document.write( "fourth term = (r)*(4*r^2)<BR>\n" );
document.write( "fourth term = 4r^3\r<BR>\n" );
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document.write( "the expression for the fourth term is 4r^3. We are given that the fourth term is also 108. Equate these two expressions and solve for r.\r<BR>\n" );
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document.write( "4r^3 = 108<BR>\n" );
document.write( "(4r^3)/4 = 108/4<BR>\n" );
document.write( "r^3 = 27<BR>\n" );
document.write( "CubeRoot(r^3) = CubeRoot(27) ... ** see note below **<BR>\n" );
document.write( "r = 3\r<BR>\n" );
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document.write( "Note: applying the cube root to both sides is the same as raising both sides to the 1/3 power. \r<BR>\n" );
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document.write( "Now that we know the common ratio is r = 3, we use it to find the missing terms we need.\r<BR>\n" );
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document.write( "Plug r = 3 into the second term equation (defined above):<BR>\n" );
document.write( "second term = 4*r<BR>\n" );
document.write( "second term = 4*3<BR>\n" );
document.write( "second term = 12\r<BR>\n" );
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document.write( "Do the same for the third term equation as well:<BR>\n" );
document.write( "third term = 4r^2<BR>\n" );
document.write( "third term = 4*(3)^2<BR>\n" );
document.write( "third term = 4*9<BR>\n" );
document.write( "third term = 36\r<BR>\n" );
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document.write( "An alternative way to compue the third term is:<BR>\n" );
document.write( "third term = (common ratio)*(second term)<BR>\n" );
document.write( "third term = (r)*(12)<BR>\n" );
document.write( "third term = (3)*(12)<BR>\n" );
document.write( "third term = 36<BR>\n" );
document.write( "and we get the same answer\r<BR>\n" );
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document.write( "Side Notes:\r<BR>\n" );
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document.write( "Plugging r = 3 into the equation for the fourth term leads to this:<BR>\n" );
document.write( "fourth term = 4r^3<BR>\n" );
document.write( "fourth term = 4*(3)^3<BR>\n" );
document.write( "fourth term = 4*27<BR>\n" );
document.write( "fourth term = 108<BR>\n" );
document.write( "which helps verify that we have the right common ratio (r).\r<BR>\n" );
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document.write( "Another way to see if we have the right common ratio is to divide each term (but the first) over its previous term. Doing so yields\r<BR>\n" );
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document.write( "(second term)/(first term) = 12/4 = 3<BR>\n" );
document.write( "(third term)/(second term) = 36/12 = 3<BR>\n" );
document.write( "(fourth term)/(third term) = 108/36 = 3\r<BR>\n" );
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document.write( "We get 3 each time. This is another way to confirm that we have the right common ratio (r).\r<BR>\n" );
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