document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=87238>Question 87238</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the geometric sequence of numbers 1, 1/3, 1/9 , 1/27… to find the following:<BR>\n" );
document.write( "a) 	What is r, the ratio between 2 consecutive terms? <BR>\n" );
document.write( "Answer:  <BR>\n" );
document.write( "Show work in this space. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b) 	Using the formula for the sum of the first n terms of a geometric sequence, what is the sum of the first 10 terms? Carry all calculations to 6 decimals on all assignments.<BR>\n" );
document.write( "Answer: <BR>\n" );
document.write( "Show work in this space.   \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "c) 	Using the formula for the sum of the first n terms of a geometric sequence, what is the sum of the first 12 terms? Carry all calculations to 6 decimals on all assignments.<BR>\n" );
document.write( "Answer:  <BR>\n" );
document.write( "Show work in this space.   \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "d) 	What observation can make about the successive partial sums of this sequence? In particular, what number does it appear that the sum will always be smaller than?<BR>\n" );
document.write( "Answer:  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #63188 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=63188\">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 to find r, simply pick any term and divide it by the previous term:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=%281%2F27%29%2F%281%2F9%29=%289%2F27%29=1%2F3 ALIGN=MIDDLE  ALT=\"r=%281%2F27%29%2F%281%2F9%29=%289%2F27%29=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Divide the term of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F27 ALIGN=MIDDLE  ALT=\"1%2F27\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F9 ALIGN=MIDDLE  ALT=\"1%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "So the ratio is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1%2F3 ALIGN=MIDDLE  ALT=\"r=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sequence is reduced by a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > each term, so the sequence is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%5En ALIGN=MIDDLE  ALT=\"%281%2F3%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281-%281%2F3%29%5E10%29%2F%281-%281%2F3%29%29 ALIGN=MIDDLE  ALT=\"S=%281-%281%2F3%29%5E10%29%2F%281-%281%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > So plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1%2F3 ALIGN=MIDDLE  ALT=\"r=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and  n=10 to find the sum of the first 10 partial sums\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281-1%2F59049%29%2F%281-%281%2F3%29%29 ALIGN=MIDDLE  ALT=\"S=%281-1%2F59049%29%2F%281-%281%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 10 power\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%2859049%2F59049-1%2F59049%29%2F%281%2F2%29 ALIGN=MIDDLE  ALT=\"S=%2859049%2F59049-1%2F59049%29%2F%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make \"1\" into an equivalent fraction with a denominator of 59049\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%2859048%2F59049%29%2F%281-%281%2F3%29%29 ALIGN=MIDDLE  ALT=\"S=%2859048%2F59049%29%2F%281-%281%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the fractions in the numerator\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%2859048%2F59049%29%2F%282%2F3%29 ALIGN=MIDDLE  ALT=\"S=%2859048%2F59049%29%2F%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the fractions in the denominator\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=177144%2F118098 ALIGN=MIDDLE  ALT=\"S=177144%2F118098\"   DISABLED_style= \"max-width:90%; height:auto;\" > Flip the 2nd fraction, multiply, and simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the sum of the first ten terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=177144%2F118098 ALIGN=MIDDLE  ALT=\"177144%2F118098\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 1.49997459736829 approximately\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "note: I chose to use fractions (to maintain accuracy), but it may be much easier for you to simply use a calculator to evaluate the sum.\r<BR>\n" );
document.write( "\n" );
document.write( "c)<BR>\n" );
document.write( "The sum of a geometric series is\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\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\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281-%281%2F3%29%5E12%29%2F%281-%281%2F3%29%29 ALIGN=MIDDLE  ALT=\"S=%281-%281%2F3%29%5E12%29%2F%281-%281%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > So plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=1%2F3 ALIGN=MIDDLE  ALT=\"r=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and  n=12 to find the sum of the first 12 partial sums\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281-1%2F531441%29%2F%281-%281%2F3%29%29 ALIGN=MIDDLE  ALT=\"S=%281-1%2F531441%29%2F%281-%281%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Raise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the 10 power\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%28531441%2F59049-1%2F59049%29%2F%281%2F2%29 ALIGN=MIDDLE  ALT=\"S=%28531441%2F59049-1%2F59049%29%2F%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make \"1\" into an equivalent fraction with a denominator of 531441\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%28531440%2F531441%29%2F%281-%281%2F3%29%29 ALIGN=MIDDLE  ALT=\"S=%28531440%2F531441%29%2F%281-%281%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the fractions in the numerator\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%28531440%2F531441%29%2F%282%2F3%29 ALIGN=MIDDLE  ALT=\"S=%28531440%2F531441%29%2F%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the fractions in the denominator\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=1594320%2F1062882 ALIGN=MIDDLE  ALT=\"S=1594320%2F1062882\"   DISABLED_style= \"max-width:90%; height:auto;\" > Flip the 2nd fraction, multiply, and simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the sum of the first twelve terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1594320%2F1062882 ALIGN=MIDDLE  ALT=\"1594320%2F1062882\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 1.49999717748537 approximately\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "d)<BR>\n" );
document.write( "It appears that the sums are approaching a finite number of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 1.5. 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 sum of the infinite series. So if we let a=1 and r=1/3 we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=1%2F%281-%281%2F3%29%29 ALIGN=MIDDLE  ALT=\"S=1%2F%281-%281%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=S=1%2F%282%2F3%29 ALIGN=MIDDLE  ALT=\"S=1%2F%282%2F3%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=3%2F2 ALIGN=MIDDLE  ALT=\"S=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >So this verifies that our series approaches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );