Problem with making explicit formula from recursive formula

Question

I'm having trouble turning this recursive formula into explicit one

$a_0 = 0,\\a_{n+1} = a_n(1 - b_n) + b_n,$

where $(b_n)$ is given sequence of real numbers.

Answer 1

Since $$\begin{align*} a_{n+1}-1=&a_n(1-b_n)-1+b_n\\=&(a_n-1)(1-b_n)\end{align*}$$

Consider instead a sequence $(c_n)$, $c_n=a_n-1$, $n=0,1,2,\ldots$,

$$\begin{align*} a_n-1=c_n=&c_{n-1}(1-b_{n-1})\\ =&c_{n-2}(1-b_{n-2})(1-b_{n-1})\\ =&c_{n-3}(1-b_{n-3})(1-b_{n-2})(1-b_{n-1})\\ &\vdots\\ =&c_{n-n}\prod_{i=1}^{n}(1-b_{n-i})\\ =&c_0\prod_{j=0}^{n-1}(1-b_j)\\ =&-\prod_{j=0}^{n-1}(1-b_j)\\ a_n=&1-\prod_{j=0}^{n-1}(1-b_j) \end{align*}$$