Three, (25 points) A chemical plant, starting from January this year, if it does not improve its production environment and continues to produce as it is, will earn 700,000 yuan per month. At the same time, it will receive penalties from the environmental protection department, with the first month's penalty being 30,000 yuan, and increasing by 20,000 yuan each subsequent month. If the plant invests 5,000,000 yuan to add recycling and purification equipment (assuming no time for equipment renovation), it can not only improve the environment but also significantly reduce raw material costs. According to estimates, the cumulative net income for the first 5 months after production is a quadratic function of production time \( n \) (in months). The cumulative income for the first 1, first 2, and first 3 months can reach 1,010,000 yuan, 2,040,000 yuan, and 3,090,000 yuan, respectively, and will stabilize at the level of the fifth month thereafter. At the same time, the plant will not only avoid penalties but also receive a one-time reward of 1,000,000 yuan from the environmental protection department. How many months will it take for the investment to start paying off, i.e., for the net income after the investment to exceed the net income without the investment?
Three, let the cumulative income for $n$ months without modifying the equipment, under the original conditions, be $a(n)$, and let the cumulative income for $n$ months after modifying the equipment be $b(n)$. From the given conditions, we have $a(n)=70 n$, and we can assume
$$
b(n)=a n^{2}+b n+c, \quad n \leqslant 5.
$$
Thus, 
$$
\left\{\begin{array}{l}
b(1)=a+b+c=101, \\
b(2)=4 a+2 b+c=204, \\
b(3)=9 a+3 b+c=309.
\end{array}\right.
$$
Solving these, we get $a=1, b=100, c=0$.
$$
\therefore b(n)=\left\{\begin{array}{ll}
n^{2}+100 n, & (n \leqslant 5) \\
b(5)+(n-5 \llbracket b(5)-b(4)], & (n>5)
\end{array}\right.
$$

That is $\square$
$$
\left.\begin{array}{l}
b(n)=\left\{\begin{array}{l}
n^{2}+100 n, \quad(n \leqslant 5) \\
109 n-20. \quad(n>5)
\end{array}\right. \\
\because b(5)-500+100=125a(n)-\left[3 n+\frac{2 n(n-1)}{2}\right](n>5)
\end{array}
\right.
$$

That is $n^{2}+41 n-420>0$.
When $n \geqslant 9$, the above inequality holds.
Therefore, the investment will start to pay off after 9 months.