document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=84140>Question 84140</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the geometric sequence of numbers 1, 1/2, 1/4, 1/8,.....to find the following:\r<BR>\n" );
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document.write( "a) What is r, the ratio between 2 consecutive terms?<BR>\n" );
document.write( "  Answer: r=1/2 divided by 1,\r<BR>\n" );
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document.write( " which is 1/2(fraction)<BR>\n" );
document.write( "Is this correct\r<BR>\n" );
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document.write( "b) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms?  Round your answer to 4 decimals.<BR>\n" );
document.write( "show work\r<BR>\n" );
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document.write( "1st term is 1X1/2=1/2<BR>\n" );
document.write( "2nd term is 1/2X1/2=1/4<BR>\n" );
document.write( "3rd term is 1/4X1/2=1/8<BR>\n" );
document.write( "4th term is 1/8X1/2=1/16<BR>\n" );
document.write( "5th term is 1/16X1/2=1/32<BR>\n" );
document.write( "6th term is 1/32X1/2=1/64<BR>\n" );
document.write( "7th term is 1/64X1/2=1/128<BR>\n" );
document.write( "8th term is 1/128X1/2=1/256<BR>\n" );
document.write( "9th term is 1/256X1/2=1/512<BR>\n" );
document.write( "10th term is 1/512X1/2=1,024\r<BR>\n" );
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document.write( "c)Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 12 terms?  Round to 4 decimals.<BR>\n" );
document.write( "show work\r<BR>\n" );
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document.write( "11th term is 1/1,024X1/2=2,048<BR>\n" );
document.write( "12 term is 1/2,048X1/2=4,096\r<BR>\n" );
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document.write( "d) What observation can you make about these sums?  In particular, what whole number does it appear that the sum will always be smaller than?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #60609 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=60609\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)<BR>\n" );
document.write( "The ratio r is the factor to get from term to term. So <BR>\n" );
document.write( "r=nth term/(n-1) term<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=%281%2F8%29%2F%281%2F4%29=%284%2F8%29=1%2F2 ALIGN=MIDDLE  ALT=\"r=%281%2F8%29%2F%281%2F4%29=%284%2F8%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<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( "So the ratio is <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;\" > (you are correct)<BR>\n" );
document.write( "The sequence is cut in half each term, so the sequence is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%5En ALIGN=MIDDLE  ALT=\"%281%2F2%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "The sum of a geometric series is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=a%281-r%5En%29%2F%281-r%29 ALIGN=MIDDLE  ALT=\"S=a%281-r%5En%29%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >where a=1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281-%281%2F2%29%5E10%29%2F%281-%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"S=%281-%281%2F2%29%5E10%29%2F%281-%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >So plug in n=10 to find the sum of the first 10 partial sums<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281-1%2F1024%29%2F%281%2F2%29 ALIGN=MIDDLE  ALT=\"S=%281-1%2F1024%29%2F%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=2046%2F1024 ALIGN=MIDDLE  ALT=\"S=2046%2F1024\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the sum of the first ten terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2046%2F1024 ALIGN=MIDDLE  ALT=\"2046%2F1024\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 1.99805 approximately\r<BR>\n" );
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document.write( "c) <BR>\n" );
document.write( "Use the same formula to find the sum of the 1st 12 terms <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=a%281-r%5En%29%2F%281-r%29 ALIGN=MIDDLE  ALT=\"S=a%281-r%5En%29%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >where a=1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281-%281%2F2%29%5E12%29%2F%281-%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"S=%281-%281%2F2%29%5E12%29%2F%281-%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >So plug in n=12 to find the sum of the first 12 partial sums<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281-1%2F4096%29%2F%281%2F2%29 ALIGN=MIDDLE  ALT=\"S=%281-1%2F4096%29%2F%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=8190%2F4096 ALIGN=MIDDLE  ALT=\"S=8190%2F4096\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the sum of the first twelve terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8190%2F4096 ALIGN=MIDDLE  ALT=\"8190%2F4096\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 1.99951 approximately\r<BR>\n" );
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document.write( "d)<BR>\n" );
document.write( "It appears that the sums are approaching a finite number of 2. This is because each term is getting smaller and smaller. This observation is justified by the fact that if <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;\" > then the infinite series will approach a finite number. In other words<BR>\n" );
document.write( "If <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;\" > (the magnitude of r has to be less than 1) then,<BR>\n" );
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;\" >Where S is the infinite series. So if we let a=1 and r=1/2 we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=1%2F%281-%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"S=1%2F%281-%281%2F2%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=S=1%2F%281%2F2%29 ALIGN=MIDDLE  ALT=\"S=1%2F%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=2 ALIGN=MIDDLE  ALT=\"S=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >So this verifies that our series approaches 2. <BR>\n" );
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