Markov chain connected with recurrent events

Question

I am reading Feller Volume 1, and this example is in page 382. I understand that $f_1= q_1$ and $f_2 = p_1 q_2$, but I don't understand how to derive $p_k$ in general (which I highlight with the purple line). I appreciate if you elaborate this.

Answer 1

Substituting $q_i=1-p_i$ for $i=1,2,\dots$ observe that:

$\begin{aligned}f_{1} & =q_{1} & & =1-p_{1}\\ f_{2} & =p_{1}q_{2} & & =p_{1}-p_{1}p_{2}\\ f_{3} & =p_{1}p_{2}q_{3} & & =p_{1}p_{2}-p_{1}p_{2}p_{3}\\ \cdots & =\cdots & & =\cdots\\ f_{k} & =p_{1}p_{2}\cdots p_{k-1}q_{k} & & =p_{1}p_{2}\cdots p_{k-1}-p_{1}p_{2}\cdots p_{k-1}p_{k} \end{aligned} $

Summation on both sides then shows that: $$f_{1}+f_{2}+f_{3}+\cdots+f_{k}=1-p_{1}p_{2}p_{3}\cdots p_{k}$$

This makes it evident that: $$p_{k}\left(1-f_{1}-f_{2}-\cdots-f_{k-1}\right)=p_{1}p_{2}p_{3}\cdots p_{k}=\left(1-f_{1}-f_{2}-\cdots-f_{k-1}-f_k\right)$$