document.write( "Question 1120309: 3,9,12,21,33,54 what are the next two numbers in the sequence
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #735968 by solver91311(24713) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Six points uniquely determine a 5th-degree polynomial function. (In general, n points uniquely determine an (n-1)th degree polynomial function).\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The general form of a 5th-degree polynomial function is:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So if

when

, then\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "We can use this result to reduce the linear system from a 6X6 system to a 5X5 system, as follows\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If

when

, then\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Taking into account the fact that

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Similarly:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

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

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Use Cramer's Rule to solve the 5X5 system. Easiest is to put your matrices into Excel and use the MDETERM function to calculate the required determinants.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "For example, your coefficient determinant and D_a determinant will be:
\n" ); document.write( "

\r\n" );
document.write( "     |    1    1    1    1    1 |\r\n" );
document.write( "     |   32   16    8    4    2 |\r\n" );
document.write( "D  = |  243   81   27    9    3 |\r\n" );
document.write( "     | 1024  256   64   16    4 |\r\n" );
document.write( "     | 3125  625  125   25    5 |\r\n" );
document.write( "\r\n" );
document.write( "     |    6    1    1    1    1 |\r\n" );
document.write( "     |    9   16    8    4    2 |\r\n" );
document.write( "D_a= |   18   81   27    9    3 |\r\n" );
document.write( "     |   30  256   64   16    4 |\r\n" );
document.write( "     |   51  625  125   25    5 |\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "and so on...\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You can also find an online matrix calculator that will do Gauss-Jordan reduction for you. The only alternative to getting the exact answer by solving the 5X5 linear system is to use a \"curve of best fit\" calculator that will handle a 5th-degree polynomial model. These generally result in inexact but very precise coefficients.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Once you have the coefficients of the 5th-degree polynomial function that models your data, you can evaluate

and

to find the next two numbers in the sequence.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "John
\n" ); document.write( "

\n" ); document.write( "My calculator said it, I believe it, that settles it
\n" ); document.write( "

\n" ); document.write( "

\n" );