7. Denote by $\langle x\rangle$ the fractional part of the real number $x$ (for instance, $\langle 3.2\rangle=0.2$ ). A positive integer $N$ is selected randomly from the set $\{1,2,3, \ldots, M\}$, with each integer having the same probability of being picked, and $\left\langle\frac{87}{303} N\right\rangle$ is calculated. This procedure is repeated $M$ times and the average value $A(M)$ is obtained. What is $\lim _{M \rightarrow \infty} A(M)$ ?
Solution: This method of picking $N$ is equivalent to uniformly randomly selecting a positive integer. Call this the average value of $\left\langle\frac{87}{303} N\right\rangle$ for $N$ a positive integer. In lowest terms, $\frac{87}{303}=\frac{29}{101}$, so the answer is the same as the average value of $\frac{0}{101}, \frac{1}{101}, \ldots, \frac{100}{101}$, which is $\frac{1+2+\cdots+100}{101 \cdot 101}=\frac{100 \cdot 101 / 2}{101 \cdot 101}=\frac{50}{101}$.