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 #843594 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=843594\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "\r\n" );
document.write( "In arithmetic progression, the general formula for the next term is\r\n" );
document.write( "\r\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;\" > = <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;\" > + d.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This equation is a recursive equation example.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In arithmetic progression, the formula for the n-th term is\r\n" );
document.write( "\r\n" );
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;\" > = <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;\" > + d*(n-1).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is an explicit formula example.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Roughly speaking, explicit formula allows calculating the term or the function explicitly.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Recursive formula allows calculating the next term of a sequence via the previous term (or several previous terms).\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "---------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you want, you may look at this youtube lesson\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "https://www.youtube.com/watch?v=nH_z9mP--c0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "about the same subject. They relate it to Algebra-I.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As more complicated/advanced popular example, there is a Fibonacci sequence.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You may read about it in this Wikipedia article \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "https://en.wikipedia.org/wiki/Fibonacci_sequence\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This sequence was originated in XII century by Italian mathematician Fibonacci <BR>\n" );
document.write( "who referred to studying a population of rabbits.<BR>\n" );
document.write( "(the true story and applications are much more rich and interesting).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For this sequence, also there are recursive and explicit formulas.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, arithmetic sequences, geometric sequences and the Fibonacci sequence are the closest examples,<BR>\n" );
document.write( "but Fibonacci is just the level of 9th - 10th grades.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "\\\\\\\\\\\\\\\\\\\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;O-o-o!  &nbsp;&nbsp;&nbsp;&nbsp;There is one more example, &nbsp;quite simple.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It is the formula for the sum of interior angles of n-sided polygon\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D ALIGN=MIDDLE  ALT=\"S%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 180*(n-2)  &nbsp;&nbsp;degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It is an explicit formula.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The recursive analogue is  &nbsp;&nbsp;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D ALIGN=MIDDLE  ALT=\"S%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn-1%5D ALIGN=MIDDLE  ALT=\"S%5Bn-1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > + 180.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The recursive formula can be deduced geometrically <BR>\n" );
document.write( "by drawing a diagonal in an n-sided polygon, &nbsp;cutting a triangle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );