If $0 \to A \to B \to C \to 0$ is a short exact sequence of vector bundles on $X$, is it necessarily true that $[{\rm Sym}^k B] = \sum_{i=0}^k [{\rm Sym}^i A \otimes {\rm Sym}^{k-i} C]$ in the Grothendieck group of vector bundles on $X$?

Is it true that ${\rm Sym}^k$ induces a map from $K_0(X)$ to $K_0(X)$?

If $0 \to A \to B \to C \to 0$ is a short exact sequence of vector bundles on $X$, is it necessarily true that $[{\rm Sym}^k B] = \sum_{i=0}^k [{\rm Sym}^i A \otimes {\rm Sym}^{k-i} C]$ in the Grothendieck group of vector bundles on $X$? (Here ${\rm Sym}^k$ stands for the $k$-th symmetric power of a vector bundle).

Is it true that ${\rm Sym}^k$ induces a map from $K_0(X)$ to $K_0(X)$?