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# Finite Differences

We know that numbers that are "finite" have *endings*. In other words, they are not "infinite," and they do not go on forever. So what exactly are "finite differences?" This is an important concept related to number sequences. Let''s find out more:

## Finite differences explained

We find finite differences by determining the sequences of forward differences. This means that we subtract the adjacent terms of a number sequence.

## Finding finite differences

For example, we might have this number sequence:

This takes the form of

_{n}= n

^{2}

We can start by taking the differences between consecutive terms. Remember, we need to subtract adjacent terms (or the numbers that are next to each other). Here''s what that looks like:

9 -4 = 5

16-9 = 7

25-16 = 9

36-25 = 11

49-36 = 13

Now all we need to do is put all of these differences together to get our "sequence of forward differences."

We can express this as:

_{n}- a

_{n-1 }= 2n+1

If the original sequence is arithmetic, we generally assume that the sequence of forward differences is constant. As we can see here, the differences increase by two with each value in the sequence.

But if the original sequence is quadratic, we generally assume that the sequence of the forward differences is arithmetic. And if the original sequence is geometric, then the sequence of forward differences is also geometric.

## Topics related to the Finite Differences

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