document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=149402>Question 149402</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do you figure out the nth term of :<BR>\n" );
document.write( "21,19,17,15...<BR>\n" );
document.write( "please help!!! xxx<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 #109615 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=109615\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Lets assume this sequence is an arithmetic sequence. The general form of the arithmetic sequence is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=d%2An%2Ba%5B1%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D=d%2An%2Ba%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the nth term, d is the difference, and <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 the first term\r<BR>\n" );
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document.write( "So lets find the difference between 2 terms (i.e. the difference between two terms)\r<BR>\n" );
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document.write( "To find the difference, simply pick any term and subtract the previous term from that selected term\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19-21=-2 ALIGN=MIDDLE  ALT=\"19-21=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Select the 2nd term (which is 19) and subtract the 1st term (which is 21) from it.\r<BR>\n" );
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document.write( "So we get a difference of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Lets pick another pair of terms to verify:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17-19=-2 ALIGN=MIDDLE  ALT=\"17-19=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Select the 3rd term (which is 17) and subtract the 2nd term (which is 19) from it.\r<BR>\n" );
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document.write( "And again, we get a difference of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "Lets pick another pair of terms to verify:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-17=-2 ALIGN=MIDDLE  ALT=\"15-17=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Select the 4th term (which is 15) and subtract the 3rd term (which is 17) from it.\r<BR>\n" );
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document.write( "And again, we get a difference of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "Since we've tested every consecutive pair of terms, we've verified that the sequence has a constant difference of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means the sequence is arithmetic\r<BR>\n" );
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document.write( "Since the difference is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-2 ALIGN=MIDDLE  ALT=\"d=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=21 ALIGN=MIDDLE  ALT=\"a%5B1%5D=21\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means the arithmetic sequence is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=-2n%2B21 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=-2n%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" >  where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > starts at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=0 ALIGN=MIDDLE  ALT=\"n=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "Notice if we plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=0 ALIGN=MIDDLE  ALT=\"n=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B0%5D=-2%280%29%2B21 ALIGN=MIDDLE  ALT=\"a%5B0%5D=-2%280%29%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=0 ALIGN=MIDDLE  ALT=\"n=0\"   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=a%5B0%5D=0%2B21 ALIGN=MIDDLE  ALT=\"a%5B0%5D=0%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B0%5D=21 ALIGN=MIDDLE  ALT=\"a%5B0%5D=21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "which is our first term\r<BR>\n" );
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document.write( "Notice if we plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=-2%281%29%2B21 ALIGN=MIDDLE  ALT=\"a%5B1%5D=-2%281%29%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   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=a%5B1%5D=-2%2B21 ALIGN=MIDDLE  ALT=\"a%5B1%5D=-2%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=19 ALIGN=MIDDLE  ALT=\"a%5B1%5D=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "which is our second term\r<BR>\n" );
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document.write( "Notice if we plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=2 ALIGN=MIDDLE  ALT=\"n=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D=-2%282%29%2B21 ALIGN=MIDDLE  ALT=\"a%5B2%5D=-2%282%29%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=2 ALIGN=MIDDLE  ALT=\"n=2\"   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=a%5B2%5D=-4%2B21 ALIGN=MIDDLE  ALT=\"a%5B2%5D=-4%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D=17 ALIGN=MIDDLE  ALT=\"a%5B2%5D=17\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "which is our third term\r<BR>\n" );
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document.write( "Notice if we plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D=-2%283%29%2B21 ALIGN=MIDDLE  ALT=\"a%5B3%5D=-2%283%29%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   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=a%5B3%5D=-6%2B21 ALIGN=MIDDLE  ALT=\"a%5B3%5D=-6%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D=15 ALIGN=MIDDLE  ALT=\"a%5B3%5D=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "which is our fourth term\r<BR>\n" );
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document.write( "Since each term corresponds to the terms of the given list, this verifies our sequence.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the list of numbers 21,19,17,15... can be generated by the sequence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=-2n%2B21 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=-2n%2B21\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > starts at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=0 ALIGN=MIDDLE  ALT=\"n=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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