document.write( "Question 1139208: What is the next number please,
\n" ); document.write( "3, -6, 12, 4, 20 \n" ); document.write( "

Algebra.Com's Answer #757045 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "The answer from the other tutor is one POSSIBLE answer that is obtained by logical reasoning.

\n" ); document.write( "But there is no single \"right\" answer to ANY problem like this, because different logical reasoning might produce a different next number.

\n" ); document.write( "And, in the end, you could put ANY number you want next and it would be a perfectly valid sequence.

\n" ); document.write( "For any problem like this to have any meaning, information must be provided indicating what kind of sequence we are looking for.

\n" ); document.write( "---------------------------------------------------------

\n" ); document.write( "Note one formal mathematical method that can theoretically be used to solve any problem like this is to assume the numbers are the values of a polynomial evaluated for x=1 to x=5.

\n" ); document.write( "5 points can be fit with a polynomial of degree 4; you could use the given values to find the polynomial that produces these numbers and then evaluate the polynomial for x=6 to find an answer that results from formal mathematics.

\n" ); document.write( "If you only need to find the next number produced by that polynomial, you can use the method of finite differences to find it, as follows:

\r\n" );
document.write( "    3   -6   12    4   20    <--  the given sequence of numbers\r\n" );
document.write( "      -9   18   -8   16      <--  the differences between terms (\"first differences\")\r\n" );
document.write( "         27   -26  24        <--  second differences\r\n" );
document.write( "           -53   50          <--  third differences\r\n" );
document.write( "              153            <--  fourth difference

\n" ); document.write( "To find the next term in the sequence assuming a polynomial of degree 4, repeat the \"153\" in the row of fourth difference and work back up the array to find the next number in the sequence:

\r\n" );
document.write( "    3   -6   12    4   20   263\r\n" );
document.write( "      -9   18   -8   16   243\r\n" );
document.write( "         27   -26  24  227\r\n" );
document.write( "           -53   50  203\r\n" );
document.write( "              153  153

\n" ); document.write( "The next number in the sequence -- assuming the sequence is produced by a polynomial of degree 4 -- is 263.
\n" ); document.write( " \n" ); document.write( "

\n" );