csat-suneung· South-Korea· csat__math-science4 marks
As shown below, a rectangle with width 6 and height 8 has a circle drawn inside with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the rectangle, creating figure $R _ { 1 }$. In figure $R _ { 1 }$, four rectangles are drawn with each segment from a vertex of the rectangle to the intersection of the diagonal and circle as the diagonal, and then a circle is drawn inside each new rectangle with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the new rectangle, creating figure $R _ { 2 }$. In each of the four congruent rectangles in figure $R _ { 2 }$, four rectangles are drawn with each segment from a vertex to the intersection of the diagonal and circle as the diagonal, and then a circle is drawn inside each new rectangle with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the new rectangle, creating figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all circles in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? (Here, the width and height of all rectangles are parallel to each other respectively.) [4 points] (1) $\frac { 37 } { 9 } \pi$ (2) $\frac { 34 } { 9 } \pi$ (3) $\frac { 31 } { 9 } \pi$ (4) $\frac { 28 } { 9 } \pi$ (5) $\frac { 25 } { 9 } \pi$
As shown below, a rectangle with width 6 and height 8 has a circle drawn inside with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the rectangle, creating figure $R _ { 1 }$.\\
In figure $R _ { 1 }$, four rectangles are drawn with each segment from a vertex of the rectangle to the intersection of the diagonal and circle as the diagonal, and then a circle is drawn inside each new rectangle with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the new rectangle, creating figure $R _ { 2 }$.\\
In each of the four congruent rectangles in figure $R _ { 2 }$, four rectangles are drawn with each segment from a vertex to the intersection of the diagonal and circle as the diagonal, and then a circle is drawn inside each new rectangle with its center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the new rectangle, creating figure $R _ { 3 }$.\\
Continuing this process, let $S _ { n }$ be the sum of the areas of all circles in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? (Here, the width and height of all rectangles are parallel to each other respectively.) [4 points]\\
(1) $\frac { 37 } { 9 } \pi$\\
(2) $\frac { 34 } { 9 } \pi$\\
(3) $\frac { 31 } { 9 } \pi$\\
(4) $\frac { 28 } { 9 } \pi$\\
(5) $\frac { 25 } { 9 } \pi$