There is a right isosceles triangle with the two legs forming the right angle each having length 1. A square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring this square is called $R _ { 1 }$. In figure $R _ { 1 }$, 2 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 2 squares is called $R _ { 2 }$. In figure $R _ { 2 }$, 4 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 4 squares is called $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points] (1) $\frac { 3 \sqrt { 2 } } { 20 }$ (2) $\frac { \sqrt { 2 } } { 5 }$ (3) $\frac { 3 } { 10 }$ (4) $\frac { \sqrt { 3 } } { 5 }$ (5) $\frac { 2 } { 5 }$
There is a right isosceles triangle with the two legs forming the right angle each having length 1. A square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring this square is called $R _ { 1 }$.\\
In figure $R _ { 1 }$, 2 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 2 squares is called $R _ { 2 }$.\\
In figure $R _ { 2 }$, 4 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 4 squares is called $R _ { 3 }$.\\
Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]\\
(1) $\frac { 3 \sqrt { 2 } } { 20 }$\\
(2) $\frac { \sqrt { 2 } } { 5 }$\\
(3) $\frac { 3 } { 10 }$\\
(4) $\frac { \sqrt { 3 } } { 5 }$\\
(5) $\frac { 2 } { 5 }$