document.write( "Question 1164340: Find the 6th term in the following sequence:
\n" ); document.write( "1, 6, 15, 28, 45 \n" ); document.write( "

Algebra.Com's Answer #788704 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Although there is an apparent mathematical pattern to the given numbers, the question is actually invalid. ANY next number would make a valid sequence.

\n" ); document.write( "ANY problem like this, in which there is no mention of what kind of sequence it is, is invalid for the same reason.

\n" ); document.write( "Nevertheless, we can assume that the problem is supposed to be solved using the apparent pattern....

\n" ); document.write( "Here is a way to show the process of finding the next number that does exactly the same thing as shown by the other tutor, but in a different way.

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document.write( "1   6  15  28  45  (given sequence)\r\n" );
document.write( "  5   9  13  17    (first differences -- differences between terms of given sequence)\r\n" );
document.write( "    4   4   4       (second differences -- differences between successive first differences)

\n" ); document.write( "The constant second differences tell us that the sequence is produced by a polynomial of degree 2. We could go ahead and find that polynomial if we wanted.

\n" ); document.write( "But to find the next number in the sequence, we only need to continue the pattern of constant second differences and then work back up the array.

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document.write( "1   6  15  28  45  66\r\n" );
document.write( "  5   9  13  17  21\r\n" );
document.write( "    4   4   4   4

\n" ); document.write( "ANSWER: The next number in the sequence (ASSUMING A SEQUENCE GENERATED BY A POLYNOMIAL) is 66.

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