decreasing random process depending on previous draw

Question

I draw a number $x_i \in [n]$ where $n \in \mathbb{N}$. I draw a number again $x_{i+1} \in [x_i]$. I repeat this drawing process till i got an 1. This generates a random sequence $(x_i)_{i=1}^k$. All draws are uniformly at random.
Question 1: What is $P[x_i = 1] $ meaning the probability of stopping at a specific index i.
Question 2: $P[x_i = k] $ where k is some number in N.
I already solved Question 2 to some recurrence $\forall k > 1$: $s_{i,k}(n) = \sum_{l=k}^n \dfrac{1}{l} s_{i-1,l}(n) = P[x_i = k] $ with base case $s_{1,l} = \dfrac{1}{n}$. But its quite a messy way and not able to be in closed form so im wondering if anyone else has a better way of solving this.