Π Product properties

Question

I would like to know why this product tends to this result: $\prod_{j=0}^{n-1}2^{n-j}=2^{\frac{n(n+1)}{2}}$. I just know that I should get out of the Π: $2^{n}$ but I don't know why $\prod_{j=0}^{n-1}2^{n-j}=2^{n^2} \prod_{j=0}^{n-1}\frac{1}{2^j}$. If someone can tell me how the term $2^{n^2}$ is out of the Π. Thanks in advance.

Answer 1

We have

$$\begin{align} \prod_{j=0}^{n-1}2^{n-j}&=\prod_{j=0}^{n-1}\dfrac{2^n}{2^j}\\&=\dfrac{2^n}{2^0}*\dfrac{2^n}{2^1}*\dfrac{2^n}{2^2}*\ldots*\dfrac{2^n}{2^{n-2}}*\dfrac{2^n}{2^{n-1}}\\&=\dfrac{ \overbrace{2^n*2^n*2^n*\ldots*2^n*2^n}^{\color{red}{n \text{ terms}}}}{2^0*2^1*2^2*\ldots*2^{n-2}*2^{n-1}}\\&=\dfrac{\overbrace{2^n*2^n*2^n*\ldots*2^n*2^n}^{\color{red}{n \text{ terms}}}}{\prod_{j=0}^{n-1}2^j}\\&=\dfrac{(2^n)^n}{\prod_{j=0}^{n-1}2^j}=\dfrac{2^{n^2}}{\prod_{j=0}^{n-1}2^j}. \end{align}$$