The numbers $(\sqrt{2} + 1), 1, (\sqrt{2} - 1)$ will be in

If ${x_1}, {x_2}, {x_3}$ as well as ${y_1}, {y_2}, {y_3}$ are in $G$.$P$. with the same common ratio,then the points $({x_1}, {y_1}), ({x_2}, {y_2})$ and $({x_3}, {y_3})$:

If $\frac{6}{3^{12}} + \frac{10}{3^{11}} + \frac{20}{3^{10}} + \frac{40}{3^{9}} + \dots + \frac{10240}{3} = 2^{n} \cdot m$,where $m$ is odd,then $m \cdot n$ is equal to

Find the fractional value of $0.125125125 \dots$

The $4^{th}$ term of a $G.P.$ is the square of its second term,and the first term is $-3$. Determine its $7^{th}$ term.

If $x = 1 + a + a^2 + \dots \infty$ $(a < 1)$ and $y = 1 + b + b^2 + \dots \infty$ $(b < 1)$,then the value of $1 + ab + a^2b^2 + \dots \infty$ is