(a) The numbers $z_{1}, z_{2}, \ldots$ satisfy

$$z_{n+1}=z_{n}+c_{n} \quad(n \geqslant 1),$$

where $c_{1}, c_{2}, \ldots$ are given constants. Find $z_{n+1}$ in terms of $c_{1}, c_{2}, \ldots, c_{n}$ and $z_{1}$.

(b) The numbers $x_{1}, x_{2}, \ldots$ satisfy

$$x_{n+1}=a_{n} x_{n}+b_{n} \quad(n \geqslant 1),$$

where $a_{1}, a_{2}, \ldots$ are given non-zero constants and $b_{1}, b_{2}, \ldots$ are given constants. Let $z_{1}=x_{1}$ and $z_{n+1}=x_{n+1} / U_{n}$, where $U_{n}=a_{1} a_{2} \cdots a_{n}$. Calculate $z_{n+1}-z_{n}$, and hence find $x_{n+1}$ in terms of $x_{1}, b_{1}, \ldots, b_{n}$ and $U_{1}, \ldots, U_{n}$.