The following appears in Fulton's Intersection Theory, and I'm not sure how to prove it.
Example 4.1.2 Assume $C$ is a purely $n$-dimensional cone, and $P(C_i)$ is not empty for each irreducible component $C_i$ of $C$. Then $$s(C) = \sum_{i \geq 0} p_* \left(c_1(\mathcal O(1))^i \cap [P(C)] \right),$$ where $p: P(C) \to X$ is the projection. (Use Lemma 1.7.2 with Proposition 4.1 (b); by Appendix B.5.2, $c_1(\mathcal O(1)) \cap P(C \oplus 1) = P(C)$.)
Lemma 1.7.2 and Propositioin 4.1 (b) both are concerned with decomposition into irreducible components, especially we need the assumption that $C$ is equidimensional, to apply Lemma 1.7.2. However, I don't even know where to apply those. Here is what I tried, which imo looks promising:
Let $i: P(C) \hookrightarrow P(C \oplus 1)$ be the inclusion, and $q: P(C \oplus 1) \to X$ be the second projection. By definition of the Segre class, we have \begin{align*} s(C) & = q_* \left( \sum_{i \geq 0} c_1(\mathcal O_{P(C \oplus 1)}(1))^i \cap [P(C \oplus 1)]\right) \\ & = q_* [P(C \oplus 1)] + q_* \left(\sum_{i \geq 1} c_1(\mathcal O_{P(C \oplus 1)}(1))^i \cap [P(C \oplus 1)]\right) \end{align*} Applying the second part of the hint to the sum we get \begin{align*} s(C) &= q_* [P(C \oplus 1)] + \sum_{i \geq 1} q_* \left( c_1(\mathcal O_{P(C \oplus 1)}(1))^{i-1} \cap i_*[P(C)] \right) \\ & \stackrel{\rho}{=} q_* [P(C \oplus 1)] + \sum_{i \geq 1} q_* i_* \left(c_1(\mathcal O_{P(C)}(1))^{i-1} \cap [P(C)]\right) \\ & = q_* [P(C \oplus 1)] + \sum_{i \geq 0} p_* \left(c_1(\mathcal O_{P(C)}(1))^{i} \cap [P(C)]\right), \end{align*} where $\rho$ is an application of the projection formula. So it remains to show $$q_* [P(C \oplus 1)] = 0.$$ Now the irreducible components of $P(C \oplus 1)$ are the $P(C_i \oplus 1)$, which have relative dimension $\geq 1$ over $X$, since each $P(C_i)$ is non-empty by assumption. This implies $q_* [P(C_i \oplus 1)] = 0$, by the definition of the push-forward of cycles, and we are done.
I'm confused because I did not use Lemma 1.7.2 and Proposition 4.1 (b), and so I didn't need the assumption that $C$ is equidimensional. Did I make a mistake?
Additional question: Is there a counterexample when $C$ is not equidimensional?
Actually you do not need Proposition 4.1 (b), but you need Lemma 1.7.2 to deduce $$c_1(\mathcal{O}(1))\cap[P(C\oplus 1)]=[P(C)].$$ Let $W_1,\dots,W_n$ be the irreducible components of $P(C\oplus 1)$ and $m_i$ be the multiplicity of $W_i$. Let $D$ denote the effective Cartier divisor $P(C)$ on $P(C\oplus 1)$. The associated line bundle $\mathcal{O}(D)$ is isomorphic to $\mathcal{O}(1)$, so we have $$ c_1(\mathcal{O}(1))\cap[P(C\oplus 1)]= \sum_im_i[D|_{W_i}] $$ by the definition of $c_1(\mathcal{O}(1))\cap({-})$. Therefore it remains to prove $$ \sum_i m_i[D|_{W_i}]=[D], $$ which is nothing but Lemma 1.7.2.