I actually have two questions on the following problem:

It is known that the leading term of the sum $\displaystyle \sum_{i=0}^n i$ is $\displaystyle n^2/2$, and for the $\displaystyle \sum_{i=0}^n i^2$ the leading term is $\displaystyle n^3/3$. Can you make a guess what's the leading term in $\displaystyle \sum_{i=0}^n i^3$? in $\displaystyle \sum_{i=0}^n i^k$? Can you prove something inductively for this?

Clearly the guess is the leading term would be: $\displaystyle n^{k+1}/{(k+1)}$

However, I have two questions here.

1) I don't really understand what this problem means by "leading term". If there are no variables in the sum, how can there be terms?

2) I couldn't find anywhere in my text if an inductive proof applies for just proving a leading term. Does it?

Thank you in advance for your help.