Three. (25 points) Write out a sequence of consecutive positive integers starting from 1, then erase one of the numbers, so that the average of the remaining numbers is $43 \frac{14}{17}$. What is the number that was erased?
Three, suppose we have written $n$ consecutive positive integers 1, 2, ..., $n$. If the number $k$ is erased, then
$$
\begin{array}{l}
\frac{(1+2+\cdots+n)-k}{n-1}=43 \frac{14}{17} \\
\Rightarrow \frac{n}{2}+\frac{n-k}{n-1}=43 \frac{14}{17} \\
\Rightarrow \frac{n}{2} \leqslant 43 \frac{14}{17} \leqslant \frac{n}{2}+1 \\
\Rightarrow 85 \frac{11}{17} \leqslant n \leqslant 87 \frac{11}{17} \Rightarrow n=86,87 .
\end{array}
$$

From equation (1), we know that $n-1$ must be a multiple of 17.
Thus, $n=86 \Rightarrow k=16$.
Therefore, the number erased is 16.
(Chen Qian, Yujing Middle School, Xishui County, Huanggang City, Hubei Province, 438200 Ma Xuemin, Xishui County Special Education School, Huanggang City, Hubei Province, 438200)