I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up.

I have a sequence of (smooth, complex, rationally connected) varieties $X_r$ of dimension $r-1$, together with a bundle $E_r$ of rank $r-1$ that comes from an excess intersection calculation. I need to compute the chern class $c_{r-1}(E_r)$ (just a number). I can write down a formula for the chern character of $E_r$ (as a function of $r$). For any fixed $r$, I can use the Newton-Girard identities and compute the chern classes, and it matches the conjecture.

In fact, using some computational data, for any class $p_{r-s}$ in $A^{r-s}(X_r)$ I can write down a (conjectural) formula for $p_{r-s}c_{s-1}(E_r)$ as a function of $r$ and $s$. I had hoped that I could use induction with the Newton-Girard identities to prove this formula (and thus the original conjecture), but I get some recursive relations that are to complicated for me to work with.

**QUESTION**
Is there any other strategy (besides using Newton-Girard identities) to compute the top chern class from chern characters that might be better when trying to compute for all $r$?