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Omni-MATH / Let S₀ be a unit square in the Cartesian plane with horizontal and vertical sides. For any n>0, the shape S_(n) is formed by adjoining 9 copies of S_(n−1) in a 3×3 grid, and then r…

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problem

Let S0S_{0} be a unit square in the Cartesian plane with horizontal and vertical sides. For any n>0n>0, the shape SnS_{n} is formed by adjoining 9 copies of Sn−1S_{n-1} in a 3×33 \times 3 grid, and then removing the center copy. Let ana_{n} be the expected value of $\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|$, where (x,y)(x, y) and $\left(x^{\prime}, y^{\prime}\right)$ are two points chosen randomly within SnS_{n}. There exist relatively prime positive integers aa and bb such that limn→∞an3n=ab\lim _{n \rightarrow \infty} \frac{a_{n}}{3^{n}}=\frac{a}{b} Compute 100a+b100 a+b.
Plain-text mathematical notation (without MathML)
Let S₀ be a unit square in the Cartesian plane with horizontal and vertical sides. For any n>0, the shape S_(n) is formed by adjoining 9 copies of S_(n−1) in a 3×3 grid, and then removing the center copy. Let a_(n) be the expected value of $\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|$, where (x,y) and $\left(x^{\prime}, y^{\prime}\right)$ are two points chosen randomly within S_(n). There exist relatively prime positive integers a and b such that lim_(n→∞)(a_(n))/(3^(n))=(a)/(b) Compute 100a+b.
Original LaTeX notation
Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \times 3$ grid, and then removing the center copy. Let $a_{n}$ be the expected value of $\left|x-x^{\prime}\right|+\left|y-y^{\prime}\right|$, where $(x, y)$ and $\left(x^{\prime}, y^{\prime}\right)$ are two points chosen randomly within $S_{n}$. There exist relatively prime positive integers $a$ and $b$ such that $$\lim _{n \rightarrow \infty} \frac{a_{n}}{3^{n}}=\frac{a}{b}$$ Compute $100 a+b$.

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Source and history

task_id
94b260c6-c774-5881-bc28-883dcdcf2909
task_key
test--94b260c6-c774-5881-bc28-883dcdcf2909
task_revision_id
1
config / split
/ test

Official source

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