# Find the next number in the sequence below?

Find the next number in the sequence below:

298 209 129 58 ?

Source: Briddles.com

The next number is

-4

because

the successive differences form an arithmetic progression:

A: 298 209 129 58 -4
B: 89 80 71 62
C: 9 9 9

• Pardon me for asking, but isn't this basically the same as fitting a second degree polynomial to three of the data points, noticing that the there's exactly one additional data point supporting the parabola hypothesis, and then extrapolating from there.
– Bass
Feb 18, 2018 at 15:28
• @Bass Kinda, yes, but I think in this case it's justified. The fact that the double-differences are the same indicates that I'm not just randomly fitting a polynomial to the data. (I also checked some other places where this puzzle is found online, and they all seem to have the same solution.) Feb 18, 2018 at 15:33
• Isn't it the defining characteristic of a second degree polynomial that the second differences are constant and non-zero?
– Bass
Feb 18, 2018 at 17:00
• The significant thing it says is that the fourth data point fits the very same second degree polynomial that was defined by the first three data points. I don't think there's any fundamental difference between fitting a polynomial of the lowest possible order and taking successive differences until all the values are identical.
– Bass
Feb 18, 2018 at 17:46
• I'm not sure I need convincing; I agree that in this case, the polynomial fit by method of differences did find a parabola that fits all four points. My complaint is just that for any three points, such a parabola exists, so just one more data point is somewhat of a meager sample size. But I guess that's a universal problem when continuing such short sequences.
– Bass
Feb 18, 2018 at 18:03

The next number in this sequence is:

$-4$

Because:

$$298 - 90 + 1 = 209$$$$209 - 80 + 0 = 128$$$$129 - 70 + (- 1) = 58$$ Notice a pattern? The recurrence relation is as follows: $$S_n=S_{n-1}-90+10n-(n-1)=S_{n-1}-89+9n$$ with $S_0=289$. Simply using this relation for the next term, $58-89+9\cdot 3=\boxed{-4}$.