If `Delta_r=|[2^(r-1)2 , 3^(r-1) , 4. 5^(r-1)] , [x , y , z] , [2^n-1 , 3^n-1 , 5^n-1]|dot` Show that `sum_(r=1)^n Delta_r=Con s t a n t`

Explanation

To solve the problem, we need to evaluate the determinant given by:

\[
\Delta_r = \begin{vmatrix}
2^{r-1} \cdot 2 & 3^{r-1} & 4 \cdot 5^{r-1} \\
x & y & z \\
2^n - 1 & 3^n - 1 & 5^n - 1
\end{vmatrix}
...