Sequences

[geodave]

The polynomial process for solving equations involves creating a set of equations based on the known numbers in the sequence.

For example, if your sequence is

1,4,10,20

You can make the following set of equations:

an³ + bn² + cn + d = f(n) for all n

a + b + c + d = 1

8a + 4b + 2c + d = 4

27a + 9b + 3c + d = 10

64a + 16b + 4c + d = 20

(You can only get as many unknowns and equations as you have numbers in the sequence.)

Now you solve by subtracting successive equations. I won't show all my work.

7a + 3b + c = 3

19a + 5b + c = 6

37a + 7b + c = 10

12a + 2b = 3

18a + 2b = 4

6a = 1

a = 1/6

b = 1/2

c = 1/3

d = 0

Therefore the next number in the sequence is:

125a + 25b + 5c + d =

125/6 + 25/2 + 5/3 + 0 =

(125 + 75 + 10) / 6 = 210/6 = 35

If you use the difference method:

1,4,10,20

3,6,10

3,4

1

1,1

3,4,5

3,6,10,15

1,4,10,20,35

You can sort of see they use the same numbers.

Now, this gets very messy very quickly, so I recommend using Excel instead.