document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203845>Question 1203845</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 20, 32, 47, 57, ?, 80<BR>\n" );
document.write( "Find the missing number of the given series </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839709 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839709\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "There is no mathematics to this; it is purely entertainment.<br><BR>\n" );
document.write( "ANY number in the missing spot makes a valid sequence.<br><BR>\n" );
document.write( "If there is no information given about what kind of sequence it is, then it is only a guessing game.<br><BR>\n" );
document.write( "Spend as much (or, better yet, as little!) time as you want trying to find a pattern that predicts the missing number -- knowing that any \"answer\" you come up with might not be \"right\".<br><BR>\n" );
document.write( "------------------------------------------------------------------<br><BR>\n" );
document.write( "Since you have received several responses saying that it's not possible to know the answer, I will add to my response to show that there is a formal mathematical way to find ONE POSSIBLE answer to the problem.<br><BR>\n" );
document.write( "The problem shows terms 1, 2, 3, 4, and 6 of a sequence.  We can find a solution using formal mathematics if we assume that the sequence is generated by a polynomial function.  If we do that, then we are looking for a polynomial f(x) for which<br><BR>\n" );
document.write( "f(1)=20; f(2)=32; f(3)=47; f(4)=57; and f(6)=80<br><BR>\n" );
document.write( "5 known function values can be fitted with a unique polynomial of degree 4, so we are looking for a function<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=ax%5E4%2Bbx%5E3%2Bcx%5E2%2Bdx%2Be ALIGN=MIDDLE  ALT=\"f%28x%29=ax%5E4%2Bbx%5E3%2Bcx%5E2%2Bdx%2Be\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "that has the 5 given function values.<br><BR>\n" );
document.write( "We can use matrices (e.g., on a graphing calculator like a TI-83 or TI-84) to find the polynomial function that generates the given function values.  Doing that gives us the following polynomial:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%285%2F12%29x%5E4-%2811%2F2%29x%5E3%2B%28289%2F12%29x%5E2-28x%2B29 ALIGN=MIDDLE  ALT=\"f%28x%29=%285%2F12%29x%5E4-%2811%2F2%29x%5E3%2B%28289%2F12%29x%5E2-28x%2B29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We can then find the missing number in the sequence by evaluating f(5), which turns out to be 64.<br><BR>\n" );
document.write( "So ONE POSSIBLE answer to the problem, using a formal mathematical process is 64.<br><BR>\n" );
document.write( "We can also find that answer, without finding the polynomial that generates the sequence, using the method of finite differences.  In a polynomial of degree 4, the 4th differences are constant.<br><BR>\n" );
document.write( "So we can call the missing term x and find the 4th differences and set them equal to find the missing term.<br><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" );
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document.write( "   20   32     47       57           x        80   given terms\r\n" );
document.write( "     12   15       10         x-57       80-x     1st differences\r\n" );
document.write( "        3     -5       x-67        137-2x        2nd differences\r\n" );
document.write( "          -8      x-62       204-3x             3rd differences\r\n" );
document.write( "             x-54     266-4x                   4th differences</PRE><BR>\n" );
document.write( "The 4th differences must be the same:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-54=266-4x ALIGN=MIDDLE  ALT=\"x-54=266-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x=320 ALIGN=MIDDLE  ALT=\"5x=320\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=64 ALIGN=MIDDLE  ALT=\"x=64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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