108. What is the maximum number of parts that $n$ lines can divide a plane?
Solution. After conducting an analysis, we establish that the number of parts will be maximal if no two of these lines are parallel and no three intersect at the same point.

Let the desired number of parts be denoted by $y_{n}$. We will construct a table:

| $n$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | $\ldots$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $y_{n}$ | 2 | 4 | 7 | 11 | 16 | 22 | 29 | 37 | $\ldots$ |

67

Let's determine the differences of this sequence!

$$
\begin{gathered}
2, \quad 4, \quad 7, \quad 11, \quad 16, \quad 22, \quad 29, \quad 37, \ldots \\
2, \quad 3, \quad 4, \quad 5, \quad 6, \quad 7, \quad 8, \ldots
\end{gathered}
$$

The numbers in row (2) form a sequence of the 1st order (an arithmetic progression with a common difference of 1). Therefore, the given sequence (1) is a sequence of the 2nd order. We can express it as a polynomial of the second degree:

$$
y_{n}=a n^{2}+b n+c
$$

Substituting $n=1,2,3$ into the equation and considering the table values, we obtain the system of equations:

$$
\left\{\begin{aligned}
a+b+c & =2 \\
4 a+2 b+c & =4 \\
9 a+3 b+c & =7
\end{aligned}\right.
$$

The solutions to this system are: $a=\frac{1}{2}, b=\frac{1}{2}$, and $c=1$.

Therefore (see (3)), $y_{n}=\frac{1}{2} n^{2}+\frac{1}{2} n+1$, or $y_{n}=\frac{1}{2}\left(n^{2}+n+2\right)$.