document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=283866>Question 283866</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Explain how the common ratio for a geometric sequence with positive terms determines whether the terms increase or decrease. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #206054 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=206054\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A geometric sequence is a sequence of the form \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar ALIGN=MIDDLE  ALT=\"ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E2 ALIGN=MIDDLE  ALT=\"ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E3 ALIGN=MIDDLE  ALT=\"ar%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ....\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "where 'a' is the starting term and 'r' is the common ratio. If 'a' is positive, then the value of 'r' will determine whether the terms will increase or decrease. If 'r' is greater than 1 (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%3E1 ALIGN=MIDDLE  ALT=\"r%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >), then the terms will increase, while on the other hand, if 'r' is less than 1 (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%3C1 ALIGN=MIDDLE  ALT=\"r%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" >), then the terms will decrease. Finally, if 'r' is equal to 1, then the terms will remain the same.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%3E1 ALIGN=MIDDLE  ALT=\"r%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then we can write 'r' as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1%2Bk ALIGN=MIDDLE  ALT=\"r=1%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E1 ALIGN=MIDDLE  ALT=\"k%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Plug this into the sequence above to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Bk%29 ALIGN=MIDDLE  ALT=\"a%281%2Bk%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Bk%29%5E2 ALIGN=MIDDLE  ALT=\"a%281%2Bk%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2Bk%29%5E3 ALIGN=MIDDLE  ALT=\"a%281%2Bk%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ....\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now expand and distribute\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bak ALIGN=MIDDLE  ALT=\"a%2Bak\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B2ak%2Bak%5E2 ALIGN=MIDDLE  ALT=\"a%2B2ak%2Bak%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B3ak%5E2%2B3ak%2Bak%5E3 ALIGN=MIDDLE  ALT=\"a%2B3ak%5E2%2B3ak%2Bak%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ....\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how the second term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bak ALIGN=MIDDLE  ALT=\"a%2Bak\"   DISABLED_style= \"max-width:90%; height:auto;\" > is bigger than 'a' since 'ak' is positive ('a' is positive and so is 'k'). Symbolically, this can be proven by showing that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3Ca%2Bak ALIGN=MIDDLE  ALT=\"a%3Ca%2Bak\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ak%3E0 ALIGN=MIDDLE  ALT=\"ak%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is true, as shown above). Similar arguments can be made to show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B2ak%2Bak%5E2 ALIGN=MIDDLE  ALT=\"a%2B2ak%2Bak%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is bigger than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bak ALIGN=MIDDLE  ALT=\"a%2Bak\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Similarly, we can show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B3ak%5E2%2B3ak%2Bak%5E3 ALIGN=MIDDLE  ALT=\"a%2B3ak%5E2%2B3ak%2Bak%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2B2ak%2Bak%5E2 ALIGN=MIDDLE  ALT=\"a%2B2ak%2Bak%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and so on.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%3C1 ALIGN=MIDDLE  ALT=\"r%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can write 'r' as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1-k ALIGN=MIDDLE  ALT=\"r=1-k\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Ck%3C1 ALIGN=MIDDLE  ALT=\"0%3Ck%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (we want to keep 'r' positive). I'll let you plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1-k ALIGN=MIDDLE  ALT=\"1-k\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 'r'. Use similar arguments shown above to show that each term will get smaller.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1 ALIGN=MIDDLE  ALT=\"r=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the sequence \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar ALIGN=MIDDLE  ALT=\"ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E2 ALIGN=MIDDLE  ALT=\"ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ar%5E3 ALIGN=MIDDLE  ALT=\"ar%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ....\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "becomes\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%29 ALIGN=MIDDLE  ALT=\"a%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%29%5E2 ALIGN=MIDDLE  ALT=\"a%281%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%29%5E3 ALIGN=MIDDLE  ALT=\"a%281%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ....\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which simplifies to \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ....\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );