document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1111290>Question 1111290</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can u please help me find the explicit formula for the series given the recursive formula ( a sub n= 3a sub (n-1)+1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #726568 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=726568\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "I was close to coming up with an explicit formula when you first posted this question; but things didn't quite seem to be working out for me.<br><BR>\n" );
document.write( "Today when I went back and looked at the problem, things fell into place.  It must have been my unconscious mind working on the problem for me....<br><br><BR>\n" );
document.write( "Let the first term be a.  Then the given recursive definition gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%281%29+=+a ALIGN=MIDDLE  ALT=\"t%281%29+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%282%29+=+3a%2B1 ALIGN=MIDDLE  ALT=\"t%282%29+=+3a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%283%29+=+9a%2B4 ALIGN=MIDDLE  ALT=\"t%283%29+=+9a%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%284%29+=+27a%2B13 ALIGN=MIDDLE  ALT=\"t%284%29+=+27a%2B13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%285%29+=+81a%2B40 ALIGN=MIDDLE  ALT=\"t%285%29+=+81a%2B40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "...<br><BR>\n" );
document.write( "In the formula for t(n), the coefficient on a is clearly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"3%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The constants in the formulas for the terms are<br><BR>\n" );
document.write( "0 1 4 13 40 ...<br><BR>\n" );
document.write( "A bit of experimentation, or perhaps some insight and logical analysis, shows the formula for this sequence to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%5E%28n-1%29-1%29%2F2 ALIGN=MIDDLE  ALT=\"%283%5E%28n-1%29-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "So, given first term a, and with the given recursive definition, the formula for the n-th term of the sequence is<br><BR>\n" );
document.write( "ANSWER:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%28n%29+=+%283%5E%28n-1%29%29a+%2B+%28%283%5E%28n-1%29%29-1%29%2F2 ALIGN=MIDDLE  ALT=\"t%28n%29+=+%283%5E%28n-1%29%29a+%2B+%28%283%5E%28n-1%29%29-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );