As suggested by Anton, you can use the O'Neill formulas in the Riemannian submersion $\mathbb C^{n+1}\to \mathbb{C} P^n$ that defines the Fubini-Study metric on $\mathbb C P^n$. This gives the following: suppose $X,Y$ are orthonormal tangent vectors at some point in $\mathbb C P^n$, and denote by $\overline X,\overline Y$ their horizontal lifts to $\mathbb C^{n+1}$ (which are also orthonormal). Then $$sec(X,Y)=1+\tfrac34\|[\overline X,\overline Y]^v\|^2=1+3|\overline g(\overline Y,J\overline X)|^2,$$ where $\overline g$ is the canonical Euclidean metric on $\mathbb C^{n+1}$, $()^v$ denotes the vertical component wrt the submersion and $J$ is the complex structure, i.e., multiplication by $\sqrt{-1}$. Note that this immediately implies that $\mathbb CP^n$ is $\tfrac14$-pinched.
With the above formula, you can easily compute the Einstein constant of $\mathbb C P^n$ to be equal to $\mu=2n+2$, see e.g. Petersen's book "Riemannian Geometry", chapter 3.
Another possible way of doing it is using that this is a Kahler manifold. The Fubini-Study metric can be thought of as $\omega_{FS}=\sqrt{-1}\partial\overline\partial\log\|z\|^2$, where $\|z\|^2$ is the square norm of a local non vanishing holomorphic section (it is independent of the choice of section by the $\partial\overline\partial$-lemma). You can then compute in local normal (holomorphic) coordinates the coefficients $g_{i\overline g_{i\bar j}$ and use that the Ricci form is given by $Ric(\omega)=-\sqrt{-1}\partial\overline\partial\log\det(g_{i\overline j})$. Ric(\omega)=-\sqrt{-1}\partial\overline\partial\log\det(g_{i\bar{j}})$. This will obviously give you the same result, but in the form$Ric(\omega_{FS})=(n+1)\omega_{FS}$. As pointed out in the comments below, the reason for the missing factor$2$in this computation is that we have to change from real orthonormal frames to complex unitary frames. 3 fixed typo. As suggested by Anton, you can use the O'Neill formulas in the Riemannian submersion$\mathbb C^{n+1}\to \mathbb{C} P^n$that defines the Fubini-Study metric on$\mathbb C P^n$. This gives the following: suppose$X,Y$are orthonormal tangent vectors at some point in$\mathbb C P^n$, and denote by$\overline X,\overline Y$their horizontal lifts to$\mathbb C^{n+1}$(which are also orthonormal). Then $$sec(X,Y)=1+\tfrac34\|[\overline X,\overline Y]\|^2=1+3|\overline Y]^v\|^2=1+3|\overline g(\overline Y,J\overline X)|^2,$$ where$\overline g$is the canonical Euclidean metric on$\mathbb C^{n+1}$,$()^v$denotes the vertical component wrt the submersion and$J$is the complex structure, i.e., multiplication by$\sqrt{-1}$. Note that this immediately implies that$\mathbb CP^n$is$\tfrac14$-pinched. With the above formula, you can easily compute the Einstein constant of$\mathbb C P^n$to be equal to$\mu=2n+2$, see e.g. Petersen's book "Riemannian Geometry", chapter 3. Another possible way of doing it is using that this is a Kahler manifold. The Fubini-Study metric can be thought of as$\omega_{FS}=\sqrt{-1}\partial\overline\partial\log\|z\|^2$, where$\|z\|^2$is the square norm of a local non vanishing holomorphic section (it is independent of the choice of section by the$\partial\overline\partial$-lemma). You can then compute in local normal (holomorphic) coordinates the coefficients$g_{i\overline j}$and use that the Ricci form is given by$Ric(\omega)=-\sqrt{-1}\partial\overline\partial\log\det(g_{i\overline j})$. This will obviously give you the same result, but in the form$Ric(\omega_{FS})=(n+1)\omega_{FS}$. As pointed out in the comments below, the reason for the missing factor$2$in this computation is that we have to change from real orthonormal frames to complex unitary frames. 2 fixed latex bug As suggested by Anton, you can use the O'Neill formulas in the Riemannian submersion$\mathbb C^{n+1}\to \mathbb{C} P^n$that defines the Fubini-Study metric on$\mathbb C P^n$. This gives the following: suppose$X,Y$are orthonormal tangent vectors at some point in$\mathbb C P^n$, and denote by$\overline X,\overline Y$their horizontal lifts to$\mathbb C^{n+1}$(which are also orthonormal). Then $$sec(X,Y)=1+\tfrac34\|[\overline X,\overline Y]\|^2=1+3|\overline g(\overline Y,J\overline X)|^2,$$ where$\overline g$is the canonical Euclidean metric on$\mathbb C^{n+1}$and$J$is the complex structure, i.e., multiplication by$\sqrt{-1}$. Note that this immediately implies that$\mathbb CP^n$is$\tfrac14$-pinched. With thisthe above formula, you can easily compute the Einstein constant of$\mathbb C P^n$to be equal to$\mu=2n+2$, see e.g. Petersen's book "Riemannian Geometry", chapter 3. Another possible way of doing it is using complex that this is a Kahler manifold. The Fubini-Study metric can be thought of as$\omega_{FS}=\sqrt{-1}\partial\overline\partial\log\|z\|^2$, where$\|z\|^2$is the square norm of a local non vanishing holomorphic section (it is independent of the choice of section by the$\partial\overline\partial$-lemma). You can then compute in local normal (holomorphic) coordinates the coefficients$g_{i\overline j}$for the metric, and computing use that the Ricci form as is given by$Ric=-\sqrt{-1}\partial\overline\partial\log\det(g_{i\overline Ric(\omega)=-\sqrt{-1}\partial\overline\partial\log\det(g_{i\overline j})$. This will obviously give you the same result, but note that in the final number will be form$n+1$, since now Ric(\omega_{FS})=(n+1)\omega_{FS}$. As pointed out in the comments below, the reason for the missing factor $2$ in this computation is in terms of that we have to change from real orthonormal frames to complex dimensionunitary frames.