document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206279>Question 1206279</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please give me a clear explanation and an example of an explicit and recursive equation Algebra 1 level. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843601 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=843601\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Recursive example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%5B1%5D+=+5%2Ca%5Bn%2B1%5D+=+a%5Bn%5D+%2B+7%29 ALIGN=MIDDLE  ALT=\"system%28a%5B1%5D+=+5%2Ca%5Bn%2B1%5D+=+a%5Bn%5D+%2B+7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The top line indicates the 1st term is 5.<BR>\n" );
document.write( "The next line says \"whatever the nth term is, add 7 to it to get the (n+1)<sup>th</sup> term\".\r<BR>\n" );
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document.write( "<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;\" > = nth term<BR>\n" );
document.write( "n is some positive integer 1,2,3,4,...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%2B1%5D ALIGN=MIDDLE  ALT=\"a%5Bn%2B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = term just after the nth term\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D+=+a%5B1%5D%2B7+=+5%2B7+=+12 ALIGN=MIDDLE  ALT=\"a%5B2%5D+=+a%5B1%5D%2B7+=+5%2B7+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the 2nd term<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=+a%5B2%5D%2B7+=+12%2B7+=+19 ALIGN=MIDDLE  ALT=\"a%5B3%5D+=+a%5B2%5D%2B7+=+12%2B7+=+19\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the 3rd term<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D+=+a%5B3%5D%2B7+=+19%2B7+=+26 ALIGN=MIDDLE  ALT=\"a%5B4%5D+=+a%5B3%5D%2B7+=+19%2B7+=+26\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the 4th term<BR>\n" );
document.write( "And so on.\r<BR>\n" );
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document.write( "We generate the sequence 5, 12, 19, 26, ...\r<BR>\n" );
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document.write( "Since we're adding the same thing to each term, we can state this sequence is arithmetic.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+5 ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > = first term<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+7 ALIGN=MIDDLE  ALT=\"d+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" > = common difference\r<BR>\n" );
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document.write( "The explicit closed form equation would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+a%5B1%5D+%2B+d%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+a%5B1%5D+%2B+d%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+5+%2B+7%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+5+%2B+7%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's try n = 4<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+5+%2B+7%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+5+%2B+7%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D+=+5+%2B+7%284-1%29 ALIGN=MIDDLE  ALT=\"a%5B4%5D+=+5+%2B+7%284-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D+=+5+%2B+7%283%29 ALIGN=MIDDLE  ALT=\"a%5B4%5D+=+5+%2B+7%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D+=+5+%2B+21 ALIGN=MIDDLE  ALT=\"a%5B4%5D+=+5+%2B+21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D+=+26 ALIGN=MIDDLE  ALT=\"a%5B4%5D+=+26\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That matches the previous <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D ALIGN=MIDDLE  ALT=\"a%5B4%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > value found through the recursive process.<BR>\n" );
document.write( "I'll let the student try other values of n. Make sure of course to only plug in positive integers.\r<BR>\n" );
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document.write( "There's pros and cons to both forms. If you wanted say the 99th term of this arithmetic sequence, then it would be tedious to follow the recursive route. <BR>\n" );
document.write( "This is because you'd need the 98th term, but the 98th term needs the 97th term, and so on.<BR>\n" );
document.write( "Use the explicit equation instead.\r<BR>\n" );
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document.write( "But the recursive method isn't always bad. It might be more intuitive to some students to see what is going on. For some students it might be easier to repeatedly add 7 to get to the desired term. The recursive form is generally practical for small values of n.<BR>\n" );
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