\( x _ { 1 } = 1 , x _ { n + 1 } = ( n + 1 ) \cdot x _ { n } , n \geq 1 \)

The given sequence is defined as follows: - \( x_1 = 1 \) - \( x_{n+1} = (n+1) \cdot x_n \) for \( n \geq 1 \) To find the general formula for the sequence, we can observe that each term is the product of the previous term and the current index. Let's calculate the first few terms of the sequence to identify a pattern: - \( x_1 = 1 \) - \( x_2 = 2 \cdot x_1 = 2 \cdot 1 = 2 \) - \( x_3 = 3 \cdot x_2 = 3 \cdot 2 = 6 \) - \( x_4 = 4 \cdot x_3 = 4 \cdot 6 = 24 \) From the calculations, we can see that the terms of the sequence are the factorials of the natural numbers. Therefore, the general formula for the sequence is: \[ x_n = n! \] So, the general formula for the sequence is \( x_n = n! \).