Chapter 7, Problem 49E

To determine

To prove: That the q-binomial theorem $\left(x+y\right)\left(x+qy\right)\left(x+q^{2}y\right)\mathrm{...}\left(x+q^{n-1}y\right)={\displaystyle \sum _{k=0}^n{\left(\begin{array}{l}n\\ k\end{array}\right)}_q{x}^{n-k}{y}^k}$, where $n{!}_q=\frac{{\displaystyle {\prod }_{j=1}^{n}1-q^j}}{{\left(1-q\right)}^n}$ is the q- factorial and ${\left(\begin{array}{l}n\\ k\end{array}\right)}_q=\frac{n{!}_q}{k{!}_q\left(n-k\right){!}_q}$ is the q-binomial coefficient.