document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=706503>Question 706503</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I calculate the next two terms for this sequence: -2,5,21,51,100<BR>\n" );
document.write( "I CAN'T see a pattern here! Which formula do I use? a+(n-1)d ? a+(n-1)d+1/2(n-1)(n-2)c? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #435267 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+Parker><B>Edwin Parker(36)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+Parker><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=435267\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <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" );
document.write( "Do you just want the next two terms? Or\r\n" );
document.write( "do you want the general formula?\r\n" );
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document.write( "Regardless of which you want we have\r\n" );
document.write( "to make a difference table:\r\n" );
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document.write( "List the terms in a row:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
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document.write( "Now subtract each number from the number \r\n" );
document.write( "just right of it to find the first differences:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
document.write( "   7   16    30    49\r\n" );
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document.write( "Now do the same with the first differences.  \r\n" );
document.write( "Subtract each number from the number just right \r\n" );
document.write( "of it to find the second differences:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
document.write( "   7   16    30    49\r\n" );
document.write( "     9    14    19\r\n" );
document.write( "   \r\n" );
document.write( "Now do the same with the second differences.  \r\n" );
document.write( "Subtract each number from the number just right \r\n" );
document.write( "of it to find the third differences:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
document.write( "   7   16    30    49\r\n" );
document.write( "     9    14    19\r\n" );
document.write( "        5     5\r\n" );
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document.write( "They are both the same, 5.  It took the third difference \r\n" );
document.write( "to get them the same.\r\n" );
document.write( "         \r\n" );
document.write( "Now if all you want are the next two terms, we just need\r\n" );
document.write( "to extend this table, by putting two more 5's\r\n" );
document.write( "beside those two and working backwards:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
document.write( "   7   16    30    49\r\n" );
document.write( "     9    14    19\r\n" );
document.write( "        5     5     5    5\r\n" );
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document.write( "Working backwards, add the first 5 to the 19 getting 24, \r\n" );
document.write( "put that to the right of the 19 above and between the\r\n" );
document.write( "5's that we added:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
document.write( "   7   16    30    49\r\n" );
document.write( "     9    14    19    24 \r\n" );
document.write( "        5     5     5    5\r\n" );
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document.write( "Still working backwards, add the second 5 to the 24 getting 29, \r\n" );
document.write( "put that to the right of the 24:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
document.write( "   7   16    30    49\r\n" );
document.write( "     9    14    19    24    29 \r\n" );
document.write( "        5     5     5    5\r\n" );
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document.write( "Still working backwards, add the second 24 to the 49 getting 73, \r\n" );
document.write( "put that to the right of the 49, above and between the 24 and 29:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
document.write( "   7   16    30    49    73\r\n" );
document.write( "     9    14    19    24     29 \r\n" );
document.write( "        5     5     5     5\r\n" );
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document.write( "Still working backwards, add the second 29 to the 73 getting 102, \r\n" );
document.write( "put that to the right of the 73:\r\n" );
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document.write( "-2   5    21    51   100\r\n" );
document.write( "   7   16    30    49    73     102\r\n" );
document.write( "     9    14    19    24     29 \r\n" );
document.write( "        5     5     5     5\r\n" );
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document.write( "Still working backwards, add the second 73 to the 100 getting 173, \r\n" );
document.write( "put that to the right of the 100, above and between the 73 and the\r\n" );
document.write( "102 (that's the next term)\r\n" );
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document.write( "-2   5    21    51   100    173\r\n" );
document.write( "   7   16    30    49    73     102\r\n" );
document.write( "     9    14    19    24     29 \r\n" );
document.write( "        5     5     5     5\r\n" );
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document.write( "Still working backwards, finally add the second 102 to the 173\r\n" );
document.write( "getting 275, \r\n" );
document.write( "put that to the right of the 173.  That's the other term\r\n" );
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document.write( "-2   5    21    51   100    173     275\r\n" );
document.write( "   7   16    30    49    73     102\r\n" );
document.write( "     9    14    19    24     29 \r\n" );
document.write( "        5     5     5     5\r\n" );
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document.write( "So the next two terms are 173 and 275\r\n" );
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document.write( "-2, 5, 21, 51, 100, 173, 275\r\n" );
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document.write( "Now to find the general term.\r\n" );
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document.write( "You mentioned these two formulas:\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "a+(n-1)d  and  a+(n-1)d+1/2(n-1)(n-2)c<BR>\n" );
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document.write( "The first formula is for when the FIRST differences are all the same.\r\n" );
document.write( "The second formula is for when the SECOND differences are all the same.\r\n" );
document.write( "These won't do since neither the first or second differences were all\r\n" );
document.write( "the same. But there is a third formula for when the THIRD differences \r\n" );
document.write( "are all the same:\r\n" );
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document.write( "a+(n-1)d+1/2!(n-1)(n-2)c+1/3!(n-1)(n-2)(n-3)b\r\n" );
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document.write( "d = the left-most number in the row of first differences\r\n" );
document.write( "c = the left-most number in the row of second differences\r\n" );
document.write( "b = the left-most number in the row of third differences\r\n" );
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document.write( "That formula can be extended as far as needed until the case\r\n" );
document.write( "where the nth differences are all the same.\r\n" );
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document.write( "So the general term here is:\r\n" );
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document.write( "-2+(n-1)(7)+(1/2)(n-1)(n-2)(9)+(1/6)(n-1)(n-2)(n-3)(5)\r\n" );
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document.write( "Multiply that all out, get it all over one denominator\r\n" );
document.write( "and you get:\r\n" );
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document.write( "(5n³-3n²+16n-30)/6\r\n" );
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document.write( "Edwin</PRE> </SPAN>\n" );
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