document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=389734>Question 389734</A>: <I><SPAN STYLE=\"word-break: break-all;\"> write a recursive formula for the sequence 15, 26, 48, 92, 180,... Then find the next term </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #276260 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=276260\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note that the difference between two consecutive terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%5D ALIGN=MIDDLE  ALT=\"a%5Bk%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bk%2B1%5D ALIGN=MIDDLE  ALT=\"a%5Bk%2B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an increasing geometric sequence. In general, we have the sequence:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+k ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D+=+k+%2B+pr%5E0+=+a%5B1%5D+%2B+pr%5E0 ALIGN=MIDDLE  ALT=\"a%5B2%5D+=+k+%2B+pr%5E0+=+a%5B1%5D+%2B+pr%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=+k+%2B+pr%5E0+%2B+pr%5E1+=+a%5B2%5D+%2B+pr%5E1 ALIGN=MIDDLE  ALT=\"a%5B3%5D+=+k+%2B+pr%5E0+%2B+pr%5E1+=+a%5B2%5D+%2B+pr%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B4%5D+=+k+%2B+pr%5E0+%2B+pr%5E1+%2B+pr%5E2+=+a%5B3%5D+%2B+pr%5E2 ALIGN=MIDDLE  ALT=\"a%5B4%5D+=+k+%2B+pr%5E0+%2B+pr%5E1+%2B+pr%5E2+=+a%5B3%5D+%2B+pr%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, etc.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It is seen that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bi%5D+=+a%5Bi-1%5D+%2B+pr%5E%28i-2%29 ALIGN=MIDDLE  ALT=\"a%5Bi%5D+=+a%5Bi-1%5D+%2B+pr%5E%28i-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i+%3E=+2 ALIGN=MIDDLE  ALT=\"i+%3E=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We can substitute p = 11 and r = 2 to obtain ,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bi%5D+=+a%5Bi-1%5D+%2B+11%2A2%5E%28i-2%29 ALIGN=MIDDLE  ALT=\"a%5Bi%5D+=+a%5Bi-1%5D+%2B+11%2A2%5E%28i-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+15 ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B5%5D+=+180 ALIGN=MIDDLE  ALT=\"a%5B5%5D+=+180\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B6%5D+=+180+%2B+11%2A2%5E4+=+356 ALIGN=MIDDLE  ALT=\"a%5B6%5D+=+180+%2B+11%2A2%5E4+=+356\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note that a recursive sequence is defined on previous terms. It is possible to write each expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bi%5D ALIGN=MIDDLE  ALT=\"a%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > in terms of i and other constants, but it wouldn't be recursive. Using the sum of a geometric sequence, we can get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bi%5D+=+15+%2B+11sum%282%5Ej%2C+j+=+0%2C+i-2%29+=+15+%2B+11%282%5E%28i-1%29+-+1%29 ALIGN=MIDDLE  ALT=\"a%5Bi%5D+=+15+%2B+11sum%282%5Ej%2C+j+=+0%2C+i-2%29+=+15+%2B+11%282%5E%28i-1%29+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Checking, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D+=+15 ALIGN=MIDDLE  ALT=\"a%5B1%5D+=+15\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B2%5D+=+15+%2B+11+=+26 ALIGN=MIDDLE  ALT=\"a%5B2%5D+=+15+%2B+11+=+26\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B3%5D+=15+%2B+11%283%29+=+48 ALIGN=MIDDLE  ALT=\"a%5B3%5D+=15+%2B+11%283%29+=+48\"   DISABLED_style= \"max-width:90%; height:auto;\" >,etc. which is the same sequence as the recursion. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );