document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=207023>Question 207023</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I was wondering if you can help me with the following problem.\r<BR>\n" );
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document.write( "I need to write the following series using the summation notation:\r<BR>\n" );
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document.write( "3-6+12-24+48-96\r<BR>\n" );
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document.write( "I can tell that you get the next number in the series by multiplying the  previous one by -2.... but I don't know where to go from there.\r<BR>\n" );
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document.write( "Please help.\r<BR>\n" );
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document.write( "Thank You </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #156522 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=156522\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, we need to find the sequence that generates the terms. You are correct in that the next term is simply the negative double of the previous term. \r<BR>\n" );
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document.write( "So we start with 3 and we multiply by -2 to get -6. Then we multiply -6 by -2 to get 12. What we're really doing is multiplying 3 by -2 twice to get to 12 (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%28-2%29%28-2%29=12 ALIGN=MIDDLE  ALT=\"%283%29%28-2%29%28-2%29=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >). To get -24, we multiply 12 by -2. We could think of it as multiplying by -2 three times like so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%28-2%29%28-2%29%28-2%29=-24 ALIGN=MIDDLE  ALT=\"%283%29%28-2%29%28-2%29%28-2%29=-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To get 48, we just multiply by 4 negative two's as such <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%28-2%29%28-2%29%28-2%29%28-2%29=48 ALIGN=MIDDLE  ALT=\"%283%29%28-2%29%28-2%29%28-2%29%28-2%29=48\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To shorten these expressions, we can use exponents. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28-2%29%5E2=12 ALIGN=MIDDLE  ALT=\"3%28-2%29%5E2=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28-2%29%5E3=-24 ALIGN=MIDDLE  ALT=\"3%28-2%29%5E3=-24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28-2%29%5E4=48 ALIGN=MIDDLE  ALT=\"3%28-2%29%5E4=48\"   DISABLED_style= \"max-width:90%; height:auto;\" > can replace the last three expressions.\r<BR>\n" );
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document.write( "To get the the nth term (some general unknown term), we need to multiply 3 by 'n' <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms. Or to simplify things, we multiply 3 by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%29%5En ALIGN=MIDDLE  ALT=\"%28-2%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the sequence is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=3%28-2%29%5En ALIGN=MIDDLE  ALT=\"a%5Bn%5D=3%28-2%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 'n' starts at n=0. Your book or your teacher may require you to start at n=1. So simply multiply by -2 to shift the entire sequence to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=3%28-2%29%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=3%28-2%29%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 'n' now starts at n=1.\r<BR>\n" );
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document.write( "To keep things simple, I'm going to use the sequence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=3%28-2%29%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=3%28-2%29%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So to get the 6th term, we plug in n=6 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B6%5D=3%28-2%29%5E%286-1%29=3%28-2%29%5E5=3%28-32%29=-96 ALIGN=MIDDLE  ALT=\"a%5B6%5D=3%28-2%29%5E%286-1%29=3%28-2%29%5E5=3%28-32%29=-96\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is indeed the 6th term (try other values of 'n' to test this sequence)\r<BR>\n" );
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document.write( "So once we've found the sequence, simply stick it in a summation symbol like so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \displaystyle \sum_{i=1}^n 3(-2)^{i-1}\">\r<BR>\n" );
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document.write( "Note: I'm using the variable 'i' (for index) as you'll commonly see this variable used in summations.\r<BR>\n" );
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document.write( "Further note: I didn't realize this, but if you want to stop the series at 96, then simply use the summation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \displaystyle \sum_{i=1}^6 3(-2)^{i-1}\">\r<BR>\n" );
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document.write( "This will simply add up the terms from 3 to 96. In other words, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \displaystyle \sum_{i=1}^6 3(-2)^{i-1}=3-6+12-24+48-96\"> </SPAN>\n" );
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