terms 1st diffs 2nd diffs 3rd diffs
-14 -1 3 -49
-15 2 -46
-12 -44
-56
Since it takes 3 differences to get to a column of
differences with all the same number, we assume
a polynomial solution of degree 3:
So we assume the general term:
an = An³+Bn²+Cn+D
a1 = -14 = A(1)³+B(1)²+C(1)+D
A+B+C+D = -14
-------------------------------
a2 = -15 = A(2)³+B(2)²+C(2)+D
8A+4B+2C+D = -15
-------------------------------
a3 = -12 = A(3)³+B(3)²+C(3)+D
27A+9B+3C+D = -12
-------------------------------
a4 = -56 = A(4)³+B(4)²+C(4)+D
64A+16B+4C+D = -56
-------------------------------
Solve this system of equations:
A+ B+ C+D = -14
8A+ 4B+2C+D = -15
27A+ 9B+3C+D = -12
64A+16B+4C+D = -56
Get:
A=-8.5, B=53, C=-100.5, D=42
General term:
an = -8.5n³+53n²-100.5n+42
-14, -15, -12, -56, -198, -489, -980, -1722, -2766, -4163, -5964, ...
Edwin
