All k-term sums of the first n natural numbers

Question

is there a formula that generates k-term sums of the first $n$ natural numbers? The numbers are not allowed to repeat. For example, for $n=10$ and $k=4$, some possible sums would be $1+2+3+4, 2+6+7+8, 5+4+8+9$, etc. Some impossible sums would be $1+2+5+8+9, 4+6, 2+2+2+2, 6+2+3+2$, etc. Pardon the lack of formatting, I am writing from my phone in the car.

Answer 1

They are, in fact, all positive integers between the minimum possible sum

$$\frac{(k)(k+1)}{2}=1+2+3+...+k$$

and the maximum possible sum

$$kn-\frac{k(k-1)}{2}=(n-k+1)+(n-k+2)+...+n.$$

Can you show, by perturbing previous sums, why all of these sums can be reached?