K-theory of projective space

Question

Is there any way to prove that the twisting sheaves $\mathcal{O}(K)$ generate the algebraic K-theory of projective space without actually using any K-theory machinery (e.g. Bott periodicity)? Like for example, just being able to write down a resolution by powers of twisting sheaves. I know that arbitrary coherent sheaves on projective space are hard to get a handle of, so it might not be possible, but it would be helpful.