Can we prove summation formula for the first $n$ terms of natural numbers through calculus? Can we prove summation formula for the first $n$ terms of natural numbers through calculus?
What about the summation of first $n$ numbers of the form $a^k$ and other summation formulas like sum of a GP or AGP?

Essentially using calculus please.

Well if you mean $a^1+a^2+...+a^n$, just plug in the arithmetic progression formula. If you want to calculate $1^k+2^k+...+n^k$, here is an iterative method that you may apply.
For instance, when we are calculating $n^3$, in fact $(n+3)(n+2)(n+1)n-(n+2)(n+1)n(n-1)=4(n+2)(n+1)n=4(n^3+3n^2+2n)$
$$\sum_{i=1}^n (n+3)(n+2)(n+1)n-(n+2)(n+1)n(n-1)=\sum_{i=1}^n 4(n^3+3n^2+2n)$$
We can subtract the sum of $n^2$ and $n$ part to get the sum of $n^3$ part.
