<PRE><B>Question 1120017</B>
&lt;br&gt;
Use the method of finite differences to find the required degree of the polynomial.  6 points define a unique polynomial of degree 5; but it is possible a polynomial of lower degree will define the given points.&lt;br&gt;&lt;pre&gt;
  0   3  16  45  96  175
    3  13  29  51  79
     10  16  22  28
        6   6   6&lt;/pre&gt;&lt;br&gt;
The third row of differences is constant; that means the points can be defined by a polynomial of degree 3:&lt;br&gt;
t(n) = an^3+bn^2+cn+d&lt;br&gt;
Substitute n = 1, 2, 3, and 4 to get four equations in the coefficients a, b, c, and d and solve the system.  Note it will always be easy to see what the next step should be in solving the system.&lt;br&gt;&lt;pre&gt;
    a +   b +  c + d = 0
   8a +  4b + 2c + d = 3
  27a +  9b + 3c + d = 16
  64a + 16b + 4c + d = 45&lt;br&gt;
Comparing successive pairs of equations...:
   7a +  3b +  c = 3
  19a +  5b +  c = 13
  37a +  7b +  c = 29&lt;br&gt;
Again comparing successive pairs of equations...:
  12a +  2b = 10
  18a +  2b = 16&lt;br&gt;
And comparing those two equations...:
   6a = 6
    a = 1&lt;br&gt;
Then back substitute the known values to find the others:
  12(1)+2b = 10
  2b = -2
  b = -1&lt;br&gt;
  7(1)+3(-1)+c = 3
  c = -1&lt;br&gt;
  1+(-1)+(-1)+d = 0
  d = 1&lt;br&gt;&lt;/pre&gt;
We have a=1, b=-1, c=-1 and d=1; the polynomial of degree 3 that defines the given numbers is&lt;br&gt;
t(n) = n^3-n^2-n+1
</PRE>