document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=619397>Question 619397</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I can do this when the numbers are whole, but fractions are really throwing me off. Thank you for any help you can provide!!\r<BR>\n" );
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document.write( "Write the geometric series -9/2+3/2-1/2+1/6-...+1/39366 in summation notation. Then using the formula for the sum of a geometric series, compute the sum. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #389644 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=389644\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are many ways to get at the answers.<BR>\n" );
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document.write( "I calculated the common ratio (the ratio of any two consecutive terms) as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=%28-1%2F2%29%2F%283%2F2%29=%28-1%2F2%29%2A%282%2F3%29=-1%2A2%2F2%2F3=-1%2F3 ALIGN=MIDDLE  ALT=\"r=%28-1%2F2%29%2F%283%2F2%29=%28-1%2F2%29%2A%282%2F3%29=-1%2A2%2F2%2F3=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28r=-1%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28r=-1%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(I chose the pair of consecutive terms so as to make my calculation so easy that that I could do in my head. It looks more complicated when I write it out, but I just asked myself what factor multiplied by 3/2 would give me -1/2, and the answer was obvious).<BR>\n" );
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document.write( "Computing that sum is not mental math, though.<BR>\n" );
document.write( "We know that the sum of the first n terms in a geometric sequence is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28a%5Bk%5D%2Ck=1%2Ck=n%29=sum%28a%5B1%5D%2Ar%5E%28k-1%29%2Ck=1%2Ck=n%29=a%5B1%5D%2A%281-r%5En%29%2F%281-r%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So for the series in the problem, substituting the values found for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D%7D%7D%2C+%7B%7B%7Br ALIGN=MIDDLE  ALT=\"a%5B1%5D%7D%7D%2C+%7B%7B%7Br\"   DISABLED_style= \"max-width:90%; height:auto;\" > and number of terms<BR>\n" );
document.write( "-9/2+3/2-1/2+1/6-...+1/39366=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28%28-9%2F2%29%2A%28-1%2F3%29%5E%28k-1%29%2Ck=1%2Ck=12%29=%28-9%2F2%29%2A+%28%281-%28-1%2F3%29%5E12%29%2F%281-%28-1%2F3%29%29%29+=+%28-3%5E2%2F2%29%2A%28%281-1%2F3%5E12%29%2F%281%2B1%2F3%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That looks ugly, let's see if we can use common denominators and simplify some powers of 3<BR>\n" );
document.write( "SUM = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%5E2%2F2%29%2A%28%28%283%5E12-1%29%2F3%5E12%29%2F%284%2F3%29%29=%28-3%5E2%2F2%29%2A%28%283%5E12-1%29%2F3%5E12%29%2A%283%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%5E2%2A%283%5E12-1%29%2A3%2F%282%2A3%5E12%2A4%29 ALIGN=MIDDLE  ALT=\"-3%5E2%2A%283%5E12-1%29%2A3%2F%282%2A3%5E12%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%5E3%2A%283%5E12-1%29%2F2%2F4%2F3%5E12=-%283%5E12-1%29%2F%288%2A3%5E9%29=-%28531441-1%29%2F%288%2A19683%29=-531440%2F157464 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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