= Solution
Because the <Levi-Civita connection> is torsion-free, its <Christoffel symbols> satisfy $\Gamma^\mu_{\rho\alpha}=\Gamma^\mu_{\alpha\rho}$. Substitute the stated coordinate expression for each term in the cyclic sum
$$
R^\mu{}_{\rho\alpha\beta}
+R^\mu{}_{\beta\rho\alpha}
+R^\mu{}_{\alpha\beta\rho}.
$$
The partial-derivative terms cancel pairwise by the lower-index symmetry of $\Gamma$, and the quadratic terms cancel after relabelling their dummy indices. Therefore
$$
\boxed{R^\mu{}_{\rho\alpha\beta}
+R^\mu{}_{\beta\rho\alpha}
+R^\mu{}_{\alpha\beta\rho}=0,}
$$
which is the <first Bianchi identity>.
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