Successive Differences of Powers

Consider the difference between the n^\text{th} and the (n-1)^\text{th} terms of the sequence a_n. a_n-a_(n-1) = a_1n^p - a_1(n-1)^p [0.4em] ⇓ [0.4em] a_n-a_(n-1) = a_1 [ n^p - (n-1)^p ] Notice that the expression in brackets is the difference of p^\text{th} powers, which can be factored as follows. x^p-y^p = (x-y)(x^(p-1)+x^(p-2)y+ ⋯ + xy^(p-2)+y^(p-1)) Using this rule, the right hand-side of the equation can be simplified. This final equation shows that taking differences reduces the degree of the polynomial by 1. If taking the differences of consecutive terms continues, after the p^\text{th} differences the degree of the polynomial will be 0. Therefore, the polynomial will be a constant.

\begin{gathered} \textbf{Conclusion} \\ \hline \\[-0.8em] \text{If a polynomial has degree }p, \text{then} \\ \text{the }p^\text{th} \text{ differences are a nonzero constant.} \end{gathered}