csat-suneung· South-Korea· csat__math-science4 marks
As shown below, there is a right isosceles triangle with the two legs forming the right angle each having length 1. Let $R _ { 1 }$ be the figure obtained by coloring a square that has 2 vertices on the hypotenuse and the remaining 2 vertices on the two legs forming the right angle. Let $R _ { 2 }$ be the figure obtained by coloring 2 congruent squares in the figure $R _ { 1 }$, where each square has 2 vertices on the hypotenuse of each of 2 congruent right isosceles triangles and the remaining 2 vertices on the two legs forming the right angle. Let $R _ { 3 }$ be the figure obtained by coloring 4 congruent squares in the figure $R _ { 2 }$, where each square has 2 vertices on the hypotenuse of each of 4 congruent right isosceles triangles and the remaining 2 vertices on the two legs forming the right angle. Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the figure $R _ { n }$ obtained at the $n$-th step. 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) $\frac { 2 } { 5 }$
As shown below, there is a right isosceles triangle with the two legs forming the right angle each having length 1. Let $R _ { 1 }$ be the figure obtained by coloring a square that has 2 vertices on the hypotenuse and the remaining 2 vertices on the two legs forming the right angle.
Let $R _ { 2 }$ be the figure obtained by coloring 2 congruent squares in the figure $R _ { 1 }$, where each square has 2 vertices on the hypotenuse of each of 2 congruent right isosceles triangles and the remaining 2 vertices on the two legs forming the right angle.
Let $R _ { 3 }$ be the figure obtained by coloring 4 congruent squares in the figure $R _ { 2 }$, where each square has 2 vertices on the hypotenuse of each of 4 congruent right isosceles triangles and the remaining 2 vertices on the two legs forming the right angle.
Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the figure $R _ { n }$ obtained at the $n$-th step. 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) $\frac { 2 } { 5 }$