SOLUTION: What is the sequence of 0,3,5,9,12,7

,,,,,

That has 6 terms, and we don't see an obvious pattern for it.
1 less than 6 is 5, so we assume a 5th degree polynomial equation 
in n for the nth term . 

That is, we assume the 5th degree polynomial



We substitute 1,2,3,4,5, and 6 for n, making 6 equations
in 6 unknowns:



Simplifying:



Write the augmented matrix:



If you are using a TI-83 or TI-84, put that
in matrix [A], then in the 2nd MATRIX MATH
menu select rref([A]} and get



You could do that without a calculator using row
operations, but it would take you a long time and
would be tedious 

Change all those decimals to fractions:



So that matrix gives us



So the equation that will produce 

,,,,,

is found by substituting those values in



 

and if you like you can find as many terms as you like by substituting
n=7,8,9,10,11,12,...:

,,,,,, etc.

Edwin