Let ${\mathrm{\Delta }}_r=\left|\begin{array}{ccc}{2}^{r-1}& 2\left(3^{r-1}\right)& 4\left(5^{r-1}\right)\\ \alpha & \beta & \gamma \\ 2^n-1& 3^n-1& 5^n-1\end{array}\right|$ , for $r=1,2,\dots ,n{\text{.}}_r$ .Then $\sum _{r=1}^n {\mathrm{\Delta }}_r$

$\sum _{r=1}^n {\mathrm{\Delta }}_r=\left|\begin{array}{ccc}a& b& c\\ \alpha & \beta & \gamma \\ 2^n-1& 3^n-1& 5^n-1\end{array}\right|$

where $a=\sum _{r=1}^n 2^{r-1}=2^n-1$

$b=\sum _{r=1}^n 2\left(3^{r-1}\right)=\frac{2\left(3^n-1\right)}{3-1}$

$=3^n-1$

and $\begin{array}{l}c=\sum _{r=1}^n 4\left(5^{r-1}\right)=\frac{4\left(5^n-1\right)}{5-1}\\ =5^n-1\end{array}$

Thus, $\sum _{r=1}^n {\mathrm{\Delta }}_r=\left|\begin{array}{ccc}a& b& c\\ \alpha & \beta & \gamma \\ a& b& c\end{array}\right|=0$