document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=84831>Question 84831</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether the following goemetric series has a FINITE sum. If a finite sum exists, find it.\r<BR>\n" );
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document.write( "1.   8+4+2+....\r<BR>\n" );
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document.write( "2.   2+3+9/2+....\r<BR>\n" );
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document.write( "Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #61119 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=61119\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1.<BR>\n" );
document.write( "It appears that this sequence is geometric, so to find the ratio between the two terms, simply divide any term you choose by the previous term. So lets pick 4, now divide that by 8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=4%2F8=1%2F2 ALIGN=MIDDLE  ALT=\"r=4%2F8=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now pick 2 and divide it by 4\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=2%2F4=1%2F2 ALIGN=MIDDLE  ALT=\"r=2%2F4=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the ratio is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and our first term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 8. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28r%29%3C1 ALIGN=MIDDLE  ALT=\"abs%28r%29%3C1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can use the formula \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=a%2F%281-r%29 ALIGN=MIDDLE  ALT=\"S=a%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the finite sum\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=8%2F%281-1%2F2%29 ALIGN=MIDDLE  ALT=\"S=8%2F%281-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=8 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1%2F2 ALIGN=MIDDLE  ALT=\"r=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=8%2F%281%2F2%29 ALIGN=MIDDLE  ALT=\"S=8%2F%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%288%2F1%29%2A%282%2F1%29 ALIGN=MIDDLE  ALT=\"S=%288%2F1%29%2A%282%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=16 ALIGN=MIDDLE  ALT=\"S=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the finite sum is 16\r<BR>\n" );
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document.write( "If you want to verify this, take a look at the following pattern\r<BR>\n" );
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document.write( "Sum of the first 2 terms <BR>\n" );
document.write( "8+4=12 <BR>\n" );
document.write( "Sum of the first 3 terms <BR>\n" );
document.write( "8+4+2=14 <BR>\n" );
document.write( "Sum of the first 4 terms <BR>\n" );
document.write( "8+4+2+1=15 <BR>\n" );
document.write( "Sum of the first 5 terms <BR>\n" );
document.write( "8+4+2+1+0.5=15.5 <BR>\n" );
document.write( "Sum of the first 6 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25=15.75 <BR>\n" );
document.write( "Sum of the first 7 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125=15.875 <BR>\n" );
document.write( "Sum of the first 8 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625=15.9375 <BR>\n" );
document.write( "Sum of the first 9 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125=15.96875 <BR>\n" );
document.write( "Sum of the first 10 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625=15.984375 <BR>\n" );
document.write( "Sum of the first 11 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125=15.9921875 <BR>\n" );
document.write( "Sum of the first 12 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625=15.99609375 <BR>\n" );
document.write( "Sum of the first 13 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125=15.998046875 <BR>\n" );
document.write( "Sum of the first 14 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625=15.9990234375 <BR>\n" );
document.write( "Sum of the first 15 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125=15.99951171875 <BR>\n" );
document.write( "Sum of the first 16 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125+0.000244140625=15.999755859375 <BR>\n" );
document.write( "Sum of the first 17 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125+0.000244140625+0.0001220703125=15.9998779296875 <BR>\n" );
document.write( "Sum of the first 18 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125+0.000244140625+0.0001220703125+6.103515625e-005=15.9999389648438 <BR>\n" );
document.write( "Sum of the first 19 terms <BR>\n" );
document.write( "8+4+2+1+0.5+0.25+0.125+0.0625+0.03125+0.015625+0.0078125+0.00390625+0.001953125+0.0009765625+0.00048828125+0.000244140625+0.0001220703125+6.103515625e-005+3.0517578125e-005=15.9999694824219 <BR>\n" );
document.write( "Sum of the first 20 terms\r<BR>\n" );
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document.write( "and you'll notice that the partial sums slowly approach 16. This verifies our answer. \r<BR>\n" );
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document.write( "-------------------------------------------------------------\r<BR>\n" );
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document.write( "2.\r<BR>\n" );
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document.write( "Once again, it looks like this sequence is geometric. So lets find the ratio r:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3%2F2 ALIGN=MIDDLE  ALT=\"r=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Pick any term (I chose 3) and divide it by the previous term 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=%289%2F2%29%2F3=%289%2F2%29%281%2F3%29=9%2F6=3%2F2 ALIGN=MIDDLE  ALT=\"r=%289%2F2%29%2F3=%289%2F2%29%281%2F3%29=9%2F6=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Pick any term (I chose <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2F2 ALIGN=MIDDLE  ALT=\"9%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and divide it by the previous term 3\r<BR>\n" );
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document.write( "So it appears that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=3%2F2 ALIGN=MIDDLE  ALT=\"r=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and our first term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. However, since <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;\" > this means our sum will not be finite. The reason why is because we keep adding on bigger and bigger numbers to our sum, which means it will grow to infinity. \r<BR>\n" );
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document.write( "Once again if you want to verify, take a look at this\r<BR>\n" );
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document.write( "Sum of the first 2 terms <BR>\n" );
document.write( "1+1.5=2.5 <BR>\n" );
document.write( "Sum of the first 3 terms <BR>\n" );
document.write( "1+1.5+2.25=4.75 <BR>\n" );
document.write( "Sum of the first 4 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375=8.125 <BR>\n" );
document.write( "Sum of the first 5 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625=13.1875 <BR>\n" );
document.write( "Sum of the first 6 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375=20.78125 <BR>\n" );
document.write( "Sum of the first 7 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625=32.171875 <BR>\n" );
document.write( "Sum of the first 8 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375=49.2578125 <BR>\n" );
document.write( "Sum of the first 9 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625=74.88671875 <BR>\n" );
document.write( "Sum of the first 10 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375=113.330078125 <BR>\n" );
document.write( "Sum of the first 11 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625=170.9951171875 <BR>\n" );
document.write( "Sum of the first 12 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625+86.49755859375=257.49267578125 <BR>\n" );
document.write( "Sum of the first 13 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625+86.49755859375+129.746337890625=387.239013671875 <BR>\n" );
document.write( "Sum of the first 14 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625+86.49755859375+129.746337890625+194.619506835938=581.858520507813 <BR>\n" );
document.write( "Sum of the first 15 terms <BR>\n" );
document.write( "1+1.5+2.25+3.375+5.0625+7.59375+11.390625+17.0859375+25.62890625+38.443359375+57.6650390625+86.49755859375+129.746337890625+194.619506835938+291.929260253906=873.787780761719 \r<BR>\n" );
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document.write( "and you can clearly see that the sums do not approach a finite number. </SPAN>\n" );
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