document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191149>Question 1191149</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the first two terms of an exponential sequence is 135 and the sum of the third and the fourth terms is 60.given that the common ratio is positive.calculate:the limit of the first n terms as n becomes larger and the least number of terms for which the 7um exceeds 240.  </SPAN>\n" );
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document.write( "Let the sequence be a, ar, ar^2, ar^3.  Then<br><BR>\n" );
document.write( "a+ar=a(1+r)=135 [1]<BR>\n" );
document.write( "a^2+ar^3=ar^2(1+r)=60 [2]<br><BR>\n" );
document.write( "Divide [2] by [1] to determine the common ratio r:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ar%5E2%281%2Br%29%29%2F%28a%281%2Br%29%29=r%5E2=60%2F135=4%2F9 ALIGN=MIDDLE  ALT=\"%28ar%5E2%281%2Br%29%29%2F%28a%281%2Br%29%29=r%5E2=60%2F135=4%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=2%2F3 ALIGN=MIDDLE  ALT=\"r=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Use r to determine the first term a.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%281%2B2%2F3%29=135 ALIGN=MIDDLE  ALT=\"a%281%2B2%2F3%29=135\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F3%29a=135 ALIGN=MIDDLE  ALT=\"%285%2F3%29a=135\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%283%2F5%29%28135%29=81 ALIGN=MIDDLE  ALT=\"a=%283%2F5%29%28135%29=81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The first term is 81; the common ratio is 2/3.<br><BR>\n" );
document.write( "The limit of the sum is (first term) divided by (1 minus the common ratio).<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=81%2F%281-2%2F3%29=81%2F%281%2F3%29=81%2A3=243 ALIGN=MIDDLE  ALT=\"S=81%2F%281-2%2F3%29=81%2F%281%2F3%29=81%2A3=243\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER 1: 243<br><BR>\n" );
document.write( "For the second question we need to use the formula for the sum of a finite number of terms of a geometric sequence.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%28a%281-r%5En%29%29%2F%281-r%29 ALIGN=MIDDLE  ALT=\"S=%28a%281-r%5En%29%29%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We want the number of terms n for which the sum is greater than 240.  Solve the equation for a sum of 240 and round up.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240=%28%2881%29%281-%282%2F3%29%5En%29%29%2F%281-%282%2F3%29%29 ALIGN=MIDDLE  ALT=\"240=%28%2881%29%281-%282%2F3%29%5En%29%29%2F%281-%282%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=80=81%281-%282%2F3%29%5En%29 ALIGN=MIDDLE  ALT=\"80=81%281-%282%2F3%29%5En%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=80=81-81%282%2F3%29%5En ALIGN=MIDDLE  ALT=\"80=81-81%282%2F3%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81%282%2F3%29%5En=1 ALIGN=MIDDLE  ALT=\"81%282%2F3%29%5En=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F3%29%5En=1%2F81 ALIGN=MIDDLE  ALT=\"%282%2F3%29%5En=1%2F81\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The variable is in an exponent -- solve using logarithms.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%282%2F3%29%5En%29=log%281%2F81%29 ALIGN=MIDDLE  ALT=\"log%28%282%2F3%29%5En%29=log%281%2F81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Alog%282%2F3%29=log%281%2F81%29 ALIGN=MIDDLE  ALT=\"n%2Alog%282%2F3%29=log%281%2F81%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=log%281%2F81%29%2Flog%282%2F3%29 ALIGN=MIDDLE  ALT=\"n=log%281%2F81%29%2Flog%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 10.838 to 3 decimal places<br><BR>\n" );
document.write( "That decimal number means with 10 terms the sum will still be less than 240 and with 11 terms it will be more than 240.<br><BR>\n" );
document.write( "ANSWER 2: 11 terms<br><BR>\n" );
document.write( " </SPAN>\n" );
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