Answer is C, i.e. Some Constant.
First Order Difference is nothing but First order derivative. Similarly, n^{th} order difference means n^{th }order Derivative.
Take polynomial f(x) of Order n :
(Where a_{i }is any Real Number and a_{0 }is NOT equal to Zero.)
First Order Difference will be : $f'(x) = a_{0}\,n\, x^{n-1} + a_1 (n-1)\, x^{n-2} + \cdot \cdot \cdot \cdot +\,\, a_{n-1}$ i.e. a polynomial of (n-1) degree.
Similarly, You can see that Second Order difference (or Derivative) will be a Polynomial of (n-2) degree. And So on, The
$n^{th}$ degree polynomial would just be a "Zero Degree" Polynomial which is :
Which is Some Constant.
We can note the following :
- The n^{th }difference, Δ^{n}, is n! a_{0}
- The n+1 difference, Δ^{n+1}, and higher differences, are all zero.
Note That :
Zero Degree Polynomial : $f(x) = a_0\,x^{0} = a_0$ (Where $a_0$ is any Real Number and can even be Zero)
One Degree Polynomial : $f(x) = a_0\,x^{1} + a_1 = a_0 \,x + a_1$ (Where $a_i$ is any Real Number and $a_0$ Can NOT be Zero)
and So on.