document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197301>Question 1197301</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write the 4 terms of the Geometric Sequence: If the sum of the first and fourth terms is 195, and the sum of the second and the third term is 60 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830526 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830526\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let's use the standard notation of a for the first term and r for the common ratio.  Then<br><BR>\n" );
document.write( "The sum of the first and fourth terms is 195:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bar%5E3=195 ALIGN=MIDDLE  ALT=\"a%2Bar%5E3=195\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Br%5E3%29=195 ALIGN=MIDDLE  ALT=\"a%281%2Br%5E3%29=195\"   DISABLED_style= \"max-width:90%; height:auto;\" >  [1]<br><BR>\n" );
document.write( "The sum of the second and third terms is 60:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%2Bar%5E2=60 ALIGN=MIDDLE  ALT=\"ar%2Bar%5E2=60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28r%29%281%2Br%29=60 ALIGN=MIDDLE  ALT=\"a%28r%29%281%2Br%29=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > [2]<br><BR>\n" );
document.write( "It should be clear that both a and r are integers, since both sums are integers.<br><BR>\n" );
document.write( "So a quick informal method of solving the problem is to look at equation [1] and find what integer r can be so that (1+r^3) is a factor of 195.<br><BR>\n" );
document.write( "r=1 --> 1+r^3=2 --> not a factor of 195<BR>\n" );
document.write( "r=2 --> 1+r^3=9 --> not a factor of 195<BR>\n" );
document.write( "r=3 --> 1+r^3=28 --> not a factor of 195<BR>\n" );
document.write( "r=4 --> 1+r^3=65 --> YES -- 195 = 3(65)<br><BR>\n" );
document.write( "So it looks as if a=3 and r=4 are the values we are looking for.  Let's check that.<br><BR>\n" );
document.write( "The sequence is 3, 12, 48, 192.  3+192 = 195, and 12+48 = 60.<br><BR>\n" );
document.write( "ANSWER: The geometric sequence is 3, 12, 48, and 192.<br><BR>\n" );
document.write( "Note that the sequence could also be 192, 48, 12, 3 -- with first term 192 and common ratio 1/4.<br><BR>\n" );
document.write( "For a formal (and ugly) algebraic solution, using equations [1] and [2] above....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%281%2Br%5E3%29%29%2F%28a%28r%29%281%2Br%29%29=195%2F60=13%2F4 ALIGN=MIDDLE  ALT=\"%28a%281%2Br%5E3%29%29%2F%28a%28r%29%281%2Br%29%29=195%2F60=13%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%281%2Br%29%281-r%2Br%5E2%29%29%2F%28a%28r%29%281%2Br%29%29=195%2F60 ALIGN=MIDDLE  ALT=\"%28a%281%2Br%29%281-r%2Br%5E2%29%29%2F%28a%28r%29%281%2Br%29%29=195%2F60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-r%2Br%5E2%29%2Fr=13%2F4 ALIGN=MIDDLE  ALT=\"%281-r%2Br%5E2%29%2Fr=13%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%281-r%2Br%5E2%29=13r ALIGN=MIDDLE  ALT=\"4%281-r%2Br%5E2%29=13r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-4r%2B4r%5E2=13r ALIGN=MIDDLE  ALT=\"4-4r%2B4r%5E2=13r\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5E2-17r%2B4=0 ALIGN=MIDDLE  ALT=\"4r%5E2-17r%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284r-1%29%28r-4%29=0 ALIGN=MIDDLE  ALT=\"%284r-1%29%28r-4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This gives us the two values we found above for the common ratio: r=4 or r=1/4.<br><BR>\n" );
document.write( "And that leads us quickly to the two sequences we found.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );