How to solve: T(n) = T(n - 1) + n

Question

I have the following worked out:

T(n) = T(n - 1) + n = O(n^2)

Now when I work this out I find that the bound is very loose. Have I done so

Answer 1

The solution is pretty easy for this one. You have to unroll the recursion:

T(n) = T(n-1) + n = T(n-2) + (n - 1) + n = 
= T(n-3) + (n-2) + (n-1) + n = ... =
= T(0) + 1 + 2 + ... + (n-1) + n 

You have arithmetic progression here and the sum is 1/2*n*(n-1). Technically you are missing the boundary condition here, but with any constant boundary condition you see that the recursion is O(n^2).