one can show that the relation between first Chern class and second Chern class of $CP^n$ is


$\frac{2(n+1)}{n} c\_2 (M)=c\_1 (M)^2$ 


here $c\_1 (M)^2=c\_1 (M)∧c\_1 (M)$.
So is there any recurrence formula for *i*-th Chern class of $CP^n$ ?



recurrence formula for *i*-th Chern class of $CP^n$
There is no need for a recurrence formula. If $h\in \mathrm H^2(\mathbb C\mathbb P^n, \mathbb Z)$ is the Poincaré dual of a hyperplane section, then $h^i$ generates $\mathrm H^{2i}(\mathbb C\mathbb P^n, \mathbb Z)$ for all $i = 0, \dots, n$. Then it follows from the Euler sequence that
$$
\mathrm c\_i(\mathbb C\mathbb P^n) = \binom{n+1}{i}h^i.
$$


