29, 11, 2, 46, 10
Subtract 29 from each term:
0, -18, -27, 17, -19
Assume we can find a, b, c, d such that:
f(x)=ax⁴+bx³+cx²+dx+29 for x≥0.
The equations below are given labels (1) to (10).
(1) -18=a+b+c+d
(2) -27=16a+8b+4c+2d
(3) 17=81a+27b+9c+3d
(4) -19=256a+64b+16c+4d
(5) (2)-2(1): 9=14a+6b+2c
(6) (3)-3(1): 71=78a+24b+6c
(7) (4)-4(1): 53=252a+60b+12c
(8) (7)-2(6): -89=96a+12b
(9) (6)-3(5): 44=36a+6b
(10) (8)-2(9): -177=24a, a=-177/24=-59/8
Substitute in the labelled equations:
(8) -89=-708+12b, b=619/12
(5) 9=-413/4+619/2+2c, 2c=-789/4, c=-789/8
(1) -18=-59/8+619/12-789/8+d, d=437/12.
f(x)=-59x⁴/8+610x³/12-789x²/8+437x/12+29
f(0)=29, f(1)=11, f(2)=2, f(3)=46, f(4)=10, f(5)=-416.
This is the best fit for the given sequence. No straight line will fit.
The sequence would continue: f(6)=-1719, f(7)=-4563, f(8)=-9789,...
Statistically, there is virtually no correlation between the figures and their position in the series, but the linear regression equation is f(x)=20.2-0.3x. This is the only line I can find which is a poor “best fit” from a statistical point of view.