Matrix and geometric sequence

The $n$ th term of a $G.P$ is given by $a_{n}=a_{1}\times r^{n-1}$

Let, $a_{1}=a$

Substituting for the terms in the determinant we get,

$\large I=\large \begin{vmatrix} g_{1}&g_{2}&\cdots&g_{\sqrt{n}} \\ g_{\sqrt{n}+1}&g_{\sqrt{n}+2}&\cdots&g_{2\sqrt{n}} \\ \vdots&\vdots&&\vdots \\ g_{n-\sqrt{n}+1}&g_{n-\sqrt{n}+2}&\cdots&g_{n} \end{vmatrix} =\large \begin{vmatrix} a&a\times r&\cdots&a\times r^{\sqrt{n}-1} \\ a\times r^{\sqrt{n}}&a\times r^{\sqrt{n}+1}&\cdots&a\times r^{2\sqrt{n}-1} \\ \vdots&\vdots&&\vdots \\ a\times r^{n-\sqrt{n}}&a\times r^{n-\sqrt{n}+1}&\cdots&a\times r^{n-1} \end{vmatrix} =r^{\sqrt{n}} \large \begin{vmatrix} a&a\times r&\cdots&a\times r^{\sqrt{n}-1} \\ a&a\times r&\cdots&a\times r^{\sqrt{n}-1} \\ \vdots&\vdots&&\vdots \\ a\times r^{n-\sqrt{n}}&a\times r^{n-\sqrt{n}+1}&\cdots&a\times r^{n-1} \end{vmatrix}$

Since Row 1 and Row 2 are identical,by properties of determinants we have

$\large I=0$