1.
Let AA be a 3×33 \times 3 matrix such that ∣adj(adj(adj A))∣=81|\text{adj}(\text{adj}(\text{adj}\,A))| = 81. If
∑n=1∞∣adj(adj A)∣n∣A∣2n+1(n2+n),\sum_{n=1}^{\infty} \frac{|\text{adj}(\text{adj}\,A)|^{n}}{|A|^{2^{n}+1}(n^{2}+n)},

then the sum is equal to: