Task 2. (10 points) Simplify the expression $2+22+222+\ldots+\underbrace{222 . . .2}_{2021}$.
Solution. Rewrite the sum $2+22+222+\ldots+\underbrace{222 \ldots .2}_{2021}$ as

$2 \cdot 1+2 \cdot 11+2 \cdot 111+\ldots+2 \cdot \underbrace{111 \ldots 1}_{2021}=2 \cdot \frac{10-1}{9}+2 \cdot \frac{10^{2}-1}{9}+2 \cdot \frac{10^{3}-1}{9}+\ldots+2 \cdot \frac{10^{2021}-1}{9}=$ and apply the formula for the geometric progression:

$$
\begin{aligned}
& =\frac{2 \cdot\left(10+10^{2}+10^{3}+\ldots+10^{2021}\right)-2 \cdot 2021}{9}=\frac{2 \cdot \frac{10 \cdot\left(10^{2021}-1\right)}{9}-2 \cdot 2021}{9}= \\
& =\frac{20 \cdot\left(10^{2021}-1\right)}{81}-\frac{2 \cdot 2021}{9}=\frac{2 \cdot 10^{2022}-36398}{81}=\frac{20 \cdot 10^{2021}-36398}{81}
\end{aligned}
$$

$$
\text { Answer. } \frac{20 \cdot\left(10^{2021}-1\right)}{81}-\frac{2 \cdot 2021}{9}=\frac{2 \cdot 10^{2022}-36398}{81}=\frac{20 \cdot 10^{2021}-36398}{81} \text {. }
$$