Question

I suppose answer is kn? But when I try drawing the tree, it looked like

So I must have done something wrong in the more detailed analysis?

Answer 1

First, your work list has length k+2 when it should probably have length k. My guess is that you meant to run from n to n-(k-1) = n-k+1.

Now if you want to sum consecutive numbers, the easiest is to remember (or derive) the formula

1 + 2 + ... + a = a(a+1)/2

Use this to figure out that the sum you’re after is

n(n+1)/2 - (n-k)(n-k+1)/2 = nk + (k-k^2)/2

as you correctly found. Now, think about big O. Since n>k, we know nk > k^2, so the latter term is really a lower term, and the whole thing is O(nk).