document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1097251>Question 1097251</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve this.     <BR>\n" );
document.write( "3 7 14 27 52 ...... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #711648 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=711648\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>There is no way to \"solve\" this, or any other problem like this.<br><BR>\n" );
document.write( "You could put any number next and it would be a valid sequence.<br><BR>\n" );
document.write( "The only way to get the \"right\" answer is to ask the author of the problem what answer he had in mind.<br><BR>\n" );
document.write( "It is always possible to find AN answer to a problem like this.  Any sequence of n terms can be fitted by a polynomial of degree (n-1).  So with your sequence of 5 numbers, there is a polynomial of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28n%29+=+an%5E4%2Bbn%5E3%2Bcn%5E2%2Bdn%2Be ALIGN=MIDDLE  ALT=\"p%28n%29+=+an%5E4%2Bbn%5E3%2Bcn%5E2%2Bdn%2Be\"   DISABLED_style= \"max-width:90%; height:auto;\" > that will produce the given sequence.<br><BR>\n" );
document.write( "To find that polynomial, you simply form a system of 5 equations in the 5 unknowns a, b, c, d, and e.  The 5 equations are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%2Bc%2Bd%2Be=3 ALIGN=MIDDLE  ALT=\"a%2Bb%2Bc%2Bd%2Be=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <-- p(1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16a%2B8b%2B4c%2B2d%2Be+=+7 ALIGN=MIDDLE  ALT=\"16a%2B8b%2B4c%2B2d%2Be+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <-- p(2)<BR>\n" );
document.write( "...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=625a%2B125b%2B25c%2B5d%2Be+=+52 ALIGN=MIDDLE  ALT=\"625a%2B125b%2B25c%2B5d%2Be+=+52\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <-- p(5)<br><BR>\n" );
document.write( "The calculations are straightforward, but very tedious.<br><BR>\n" );
document.write( "Or a graphing calculator with matrix capability can be used to make the work relatively easy.<br><BR>\n" );
document.write( "But the answer obtained using the degree 4 polynomial might not be the \"right\" one that the author of the problem wanted........<br><br><BR>\n" );
document.write( "I see that another tutor has already supplied an answer that is very similar to one I was going to add to my response....<br><BR>\n" );
document.write( "If you notice that the numbers are \"approximately doubled\" each time, you can figure out a pattern:<BR>\n" );
document.write( "3*2 plus 1 gives you 7<BR>\n" );
document.write( "7*2 plus 0 gives you 14<BR>\n" );
document.write( "14*2 plus -1 gives you 27<BR>\n" );
document.write( "27*2 plus -2 gives you 52<br><BR>\n" );
document.write( "To get an explicit formula for the n-th term in the sequence, note that the first term is 3 and each term after that is approximately 2 times the previous term.  Then the explicit formula is going to be something of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A2%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"3%2A2%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The values produced by that formula alone are<BR>\n" );
document.write( " 3, 6, 12, 24, 48, ...<br><BR>\n" );
document.write( "Comparing those values to the actual values<BR>\n" );
document.write( " 3, 7, 14, 27, 52, ...<BR>\n" );
document.write( "we see that an explicit formula for the n-th term in the sequence is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A2%5E%28n-1%29%2B%28n-1%29 ALIGN=MIDDLE  ALT=\"3%2A2%5E%28n-1%29%2B%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><BR>\n" );
document.write( "Finally, note that the sequence you get from the polynomial method I described first will be different from the one produced by this other method.<br><BR>\n" );
document.write( "And while both sequences would be valid answers to the problem, it is far more likely that the intended answer is the one obtained by this second method. </SPAN>\n" );
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