1801. The production of industrial products $(Y)$ by a certain enterprise over the years of a seven-year plan $(X)$ is characterized by the following data:

| $X$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $Y$, million rubles | 0.5 | 0.5 | 1.5 | 3.5 | 6.5 | 10.5 | 15.5 |

Align the dependence of $Y$ on $X$ with the parabola $y=a x^{2}+b x+c$.
Solution. To find the parameters $a, b, c$ using the method of least squares, it is necessary to solve system (4), in which in this case $x_{i}$ is the year of the seven-year period; $y_{i}$ is the corresponding production. The coefficients of system (4) are calculated according to the following scheme:

| $x_{i}$ | $y_{i}$ | $x_{i}^{2}$ | $x_{i}^{3}$ | $x_{i}^{4}$ | $x_{i} y_{i}$ | $x_{i}^{2} y_{i}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 0.5 | 1 | 1 | 1 | 0.5 | 0.5 |
| 2 | 0.5 | 4 | 8 | 16 | 1 | 2 |
| 3 | 1.5 | 9 | 27 | 81 | 4.5 | 13.5 |
| 4 | 3.5 | 16 | 64 | 256 | 14 | 56 |
| 5 | 6.5 | 25 | 125 | 625 | 32.5 | 162.5 |
| 6 | 10.5 | 36 | 216 | 1296 | 63 | 378 |
| 7 | 15.5 | 49 | 343 | 2401 | 108.5 | 159.5 |
| Sum 28 | 38.5 | 140 | 784 | 4676 | 224 | 1372 |

We form the system to determine $a, b, c$ :

$$
\begin{aligned}
4676 a+784 b+140 c & =1372 \\
784 a+140 b+28 c & =224 \\
140 a+28 b+7 c & =38.5
\end{aligned}
$$

From here, $a=0.5 ; b=-1.5 ; c=1.5$.