document.write( "Question 1151348: Construct a difference table to predict the next term of the sequence
\n" ); document.write( "-1,4,21,53,103,174,269,... \n" ); document.write( "

Algebra.Com's Answer #773082 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "A table showing the original sequence and the rows of first, second, and third differences:

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document.write( "-1    4   21   53  103  174  269\r\n" );
document.write( "    5   17   32  50   71   95\r\n" );
document.write( "      12   15  18   21   24\r\n" );
document.write( "         3    3   3    3

\n" ); document.write( "There is a constant difference of 3 in the row of third differences. To find the next term in the sequence, place another difference of 3 in the third row and work back up the array:

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document.write( "-1    4   21   53  103  174  269   391\r\n" );
document.write( "    5   17   32  50   71   95   122\r\n" );
document.write( "      12   15  18   21   24   27\r\n" );
document.write( "         3    3   3    3    3

\n" ); document.write( "The predicted next term of the sequence is 391.

\n" ); document.write( "By the way.... The constant difference in the row of third differences means the sequence is generated by a polynomial of degree 3. The coefficient of the leading term is the constant difference (3) divided by the factorial of the degree of the polynomial (3!=6). So the coefficient of the leading term is

\n" ); document.write( " $\"%283%2F%283%21%29%29+=+3%2F6+=+%281%2F2%29\"$

\n" ); document.write( "and the leading term of the polynomial is then

\n" ); document.write( " $\"%281%2F2%29x%5E3\"$
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