Solution
= Solution
The mixed $SU(N)^2U(1)_R$ anomaly receives $I(\mathrm{adj})=2N$ from the gaugino. A chiral multiplet of R-charge $r$ contains a fermion of charge $r-1$. With
$$
R[S]=\frac{2-N}{N+2},
\qquad R[\widetilde\Phi]=1,
$$
the symmetric-tensor fermion contributes
$$
I(S)(R[S]-1)
=(N+2)\left(\frac{2-N}{N+2}-1\right)=-2N,
$$
while every antifundamental fermion contributes zero. The total is $2N-2N=0$, so this is a nonanomalous <R-symmetry> of the quantum theory.