document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=708337>Question 708337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Construct a difference table to predict the next term of the sequence.<BR>\n" );
document.write( "3, &#8722;2, 0, 15, 49, 108, . . . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #436358 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=436358\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Too late probably, but here's what I would do.<BR>\n" );
document.write( "I would put those numbers in a column, like this:<BR>\n" );
document.write( "3<BR>\n" );
document.write( "-2<BR>\n" );
document.write( "0<BR>\n" );
document.write( "15<BR>\n" );
document.write( "49<BR>\n" );
document.write( "108<BR>\n" );
document.write( "I would make a second column with the differences.<BR>\n" );
document.write( "subtract the first number from the second, the second number from the third, and so on.<BR>\n" );
document.write( "-2-3=-5<BR>\n" );
document.write( "0-(-2)=2<BR>\n" );
document.write( "15-0=15<BR>\n" );
document.write( "49-15=34<BR>\n" );
document.write( "108-49=59<BR>\n" );
document.write( "Since I see no pattern in those differences,<BR>\n" );
document.write( "I would make a third column with the differences of those differences:<BR>\n" );
document.write( "2-(-5)=7<BR>\n" );
document.write( "15-2=13<BR>\n" );
document.write( "34-15=19<BR>\n" );
document.write( "59-34=25<BR>\n" );
document.write( "Then I would calculate differences of differences of differences for a fourth column:<BR>\n" );
document.write( "13-7=6<BR>\n" );
document.write( "19-13=6<BR>\n" );
document.write( "25-19=6<BR>\n" );
document.write( "It could look like this:<BR>\n" );
document.write( "...3...-5....7...6<BR>\n" );
document.write( "..-2....2...13...6<BR>\n" );
document.write( "...0...15...19...6<BR>\n" );
document.write( "..15...34...25<BR>\n" );
document.write( "..49...59<BR>\n" );
document.write( ".108<BR>\n" );
document.write( "Finally, I see a pattern.<BR>\n" );
document.write( "All the numbers in the fourth column are 6. there would be more 6's if we continue the pattern.<BR>\n" );
document.write( "The numbers on the third column were increasing by 6 as they went down.<BR>\n" );
document.write( "The number after 25 would have been 25+6=31.<BR>\n" );
document.write( "The numbers in the second column increased by adding the number from the third column.<BR>\n" );
document.write( "-5+7=2<BR>\n" );
document.write( "2+15=15<BR>\n" );
document.write( "15+19=34<BR>\n" );
document.write( "34+25=59<BR>\n" );
document.write( "That means that the number after 59 would be 59+31=90.<BR>\n" );
document.write( "the numbers in the first column were calculated the same way using the second column:<BR>\n" );
document.write( "15+34=49<BR>\n" );
document.write( "49+59=108<BR>\n" );
document.write( "What comes next is<BR>\n" );
document.write( "108+90=198<BR>\n" );
document.write( "The next number in the sequence is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28198%29 ALIGN=MIDDLE  ALT=\"highlight%28198%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The table would be extended like this:<BR>\n" );
document.write( "...3...-5....7...6<BR>\n" );
document.write( "..-2....2...13...6<BR>\n" );
document.write( "...0...15...19...6<BR>\n" );
document.write( "..15...34...25...6<BR>\n" );
document.write( "..49...59...31...6<BR>\n" );
document.write( ".108...90...37<BR>\n" );
document.write( ".198..127<BR>\n" );
document.write( ".325 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );