---

Problem 959

---

Asymmetric Random Walk

A frog is placed on the number line. Every step the frog jumps either $a$ units to the left or $b$ units to the right, both with $1/2$ probability.

Define $f(a, b)$ as the limit $\lim_{n \to \infty} \frac{c_n}n$ where $c_n$ is the expected number of unique numbers visited in the first $n$ steps. You are given $f(1, 1) = 0$ and $f(1, 2) \approx 0.427050983$.

Find $f(89, 97)$. Give your answer rounded to nine digits after the decimal point.

---

非对称随机游走

一只青蛙被放置在数轴上；每一步，青蛙要么向左跳$a$个单位，要么向右跳$b$个单位，两种情况的概率均为$1/2$。

定义$f(a, b)$为极限$\lim_{n \to \infty} \frac{c_n}n$，其中$c_n$是前$n$步中青蛙落过的不同整数的期望数量。已知$f(1, 1) = 0$，$f(1, 2) \approx 0.427050983$。

求$f(89, 97)$，并四舍五入到小数点后九位作为你的答案。

---