Problem Analysis #common_16

Recorded Python call:

# Compute a_3 for the sequence a_1 = 1, a_{n+1} = n^2 * a_n + 1
a = 1
for n in range(1, 3):
    a = n**2 * a + 1
    print(f"n={n}: a_{n+1} = {a}")
print(f"Final a_3 = {a}")

The sequence is defined by $a_1 = 1$ and $a_{n+1} = n^2 a_n + 1$.

• For $n = 1$: $a_2 = 1^2 \cdot a_1 + 1 = 1 \cdot 1 + 1 = 2$.
• For $n = 2$: $a_3 = 2^2 \cdot a_2 + 1 = 4 \cdot 2 + 1 = 9$.

Thus, $a_3 = 9$.

$\boxed{9}$