Question 968243
<pre>
3, -2,  1, 18, 55, 118, 213

List them in a column, then form the differences to the right by
subtracting each number from the one just below it

  3  -5   8  6
 -2   3  14  6
  1  17  20  6
 18  37  26  6
 55  63  32
118  95
213

So we see that the 3rd differences are all 6's.

So we assume a 3rd degree polynomial general term:

{{{An^3+Bn^2+Cn+D=a[n]}}}

There are 4 unknowns A,B,C,D so we will only need
the first 4 terms.

We substitute n=1 and a<sub>1</sub>=3
We substitute n=2 and a<sub>2</sub>=-2
We substitute n=3 and a<sub>3</sub>=1
We substitute n=4 and a<sub>4</sub>=18

So we have the system:

{{{system(A+B+C+D=3,8A+4B+2C+D=-2,27A+9B+3C+D=1,
64A+16B+4C+D=18)}}}

Solve by any method and get A=1, B=-2, C=-6, D=10

So the general equation is

{{{a[n]=n^3-2n^2-6n+10}}}

Edwin</pre>