Sequences

[geodave]

You have to take the difference of every two numbers, so each list has one less number:

1, 2, 3, 4, 5, 126, 127

1,1,1,1,121,1

0,0,0,120,-120

0,0,120,-240

0,120,-360

120,-480

-600

(then add the last number again)

-600,-600

(now move back up the lists, adding a new number to the end)

120,-480,-1080

0,120,-360,-1440

0,0,120,-240,-1680

0,0,0,120,-120,-1800

1,1,1,1,121,1,-1799

Now, this can seem very counter-intuitive. However, if you use the polynomial method you'll get a similar result. This is a high-order equation which eventually dives very negative. That's probably the only way you get a curve that hits 1 all those times with 121 in the middle.

It's a lot more common to use this method for increasing sequences:

1,3,6,10,15

2,3,4,5

1,1,1

1,1,1,1 (you can go back up once all the numbers are the same)

2,3,4,5,6

1,3,6,10,15,21

I'll make a separate posting on the polynomial method.