How can I calculate this repeating power?

Question

I'm considering the stochastic tree. When root node is activated, the $n$ child nodes get the signal. But the probability of activation is $p$. $d$ is index of layer and starts from 1 that is indicating the first child layer. The activated nodes of layer $d$ is called $a_d$. The probability that a layer $d$ has no activated nodes is $$ P(a_d=0) = q^n(r(r(r(...r(r+1)^n...+1)^n+1)^n+1)^n+1)^n $$ where $q=1-p$ and $r=pq^{n-1}$. The $n$ power related to $r$ is repeated $d$ times. What I want to know is the analytic expression of $P(a_d=0)$ or at least proving, $$ \lim_{d \rightarrow \infty} P(a_d=0)=1 $$ I conducted numerical calculation and what I got is that when $np < 1$ the sequence is converge to 1 very fast($d=10^1$). When $np = 1$, converge speed is decreased($d=10^5$). When $np > 1$, the sequence seems not converge to 1($d=10^6$, the value was 0.16).