The following is a problem from my textbook:
Using the telescoping method, show that $\sum\limits_{k=1}^n k^{p}$ can always be written in the form $\dfrac{n^{p+1}}{p+1} + An^{p} + Bn^{p-1}+Cn^{p-2} + \cdots$
The following is the answer from the solutions manual:
Why do we have to assume that the statement is true for all natural numbers $\le p$? What if we assumed that the statement is true for natural numbers less than $p$ and greater than or equal to $2$, since the statement is $S(r):=\sum\limits_{k=1}^n k^{r}$ can always be written in the form $$\frac{n^{r+1}}{r+1} + An^{r} + Bn^{r-1}+Cn^{r-2} + \cdots$$