Gabeland

Let’s say we have a sequence, and we want to guess what polynomial generated it.
For example:
3, 10, 19, 30, 43, 58, 75, …
Finite differences helps with determining that.

How it works is, take the difference between each pair of terms.
10 – 3 = 7, then 19 – 10 = 9, then 30 – 19 = 11, etc.
The new sequence is 7, 9, 11, 13, 15, 17, …
That’s the odd numbers! Now, which sequence has consecutive pairs of terms differing by odd numbers? The squares. Except this clearly isn’t the squares. So let’s subtract it off:
3 – 1 = 2
10 – 4 = 6
19 – 9 = 10
etc.
2, 6, 10, 14, 18, 22, 26, … is the result.
That’s 4n – 2. So, the function of the original sequence is n² + 4n – 2.

You can use it for more general cases too.
Let’s try 8, 33, 88, 185, 336, 553, 848.

I’m not showing each individual subtraction this time, but here’s the result:
Step 1: 25, 55, 97, 151, 217, 295
Step 2: 30, 42, 54, 66, 78
Step 3: 12, 12, 12, 12

At step 3, every term is the same. That means that there has to be a cube term, and to find the coefficient, divide the constant term by 3!, or 6. That means there’s a 2n³ term. Let’s subtract that off.
6, 17, 34, 57, 86, 121, 162

It still isn’t clear what polynomial this is. But now we can repeat the process with this new sequence.
Step 1: 11, 17, 23, 29, 35, 41
Step 2: 6, 6, 6, 6, 6

Now, every term is the same by step 2. That means there’s a square term, and we can divide the constant term by 2!, which is 2, giving us 3n². Subtract that off again.
3, 5, 7, 9, 11, 13, 15
You could do it again, and get a constant term by step 1, but I’m sure you recognize this as 2n + 1.
So, the full function is 2n³ + 3n² + 2n + 1.

Finite differences also works for some other functions.
Take this sequence: 3, 8, 17, 32, 57, 100, 177, …
Step 1: 5, 9, 15, 25, 43, 77
Step 2: 4, 6, 10, 18, 34
Step 3: 2, 4, 8, 16
That’s the powers of 2. That means that 2ⁿ must be a term in the original function. In fact, the original function was 2ⁿ + n².