Can we tell the second, third, or any order differences from the first-order differences?
No, see solution.
Practice makes perfect
Let's see what we can and cannot infer from the first-order difference sequence.
We cannot tell what f(1) is. Only knowing the difference of the numbers, it is not possible to tell the numbers themselves.
However, if someone tells us f(1) then by knowing the first-order difference sequence we can tell f(2), f(3), and so on.
The first-order differences tell us the function up to a constant.
We should be able to find the degree.
Let's see how.
If the first-order difference sequence is constant, the function is linear.
If the first-order difference sequence is not constant, we check the second-order differences (the difference of the differences). If this is constant, the function is quadratic.
If neither the first- nor the second-order differences are constants, we check the third-order differences and keep going until we find a constant sequence.
If the function is a polynomial eventually we will find a constant sequence, so we can tell the degree of the function.
