PROBLEM 1. Calculate the sum $S_{n}=1+12+112+\ldots+\underbrace{111 \ldots 1}_{(n-1) \text { times }} 2$, for $n \geq 2$.
## PROBLEM 1.

$(2 p)$

$$
S_{n}=1+(11+1)+(111+1)+\ldots+(\underbrace{111 \ldots 1}_{n \text { ones }}+1)
$$

$(1 p)$

$$
S_{n}=(1+11+111+\ldots+\underbrace{111 \ldots 1}_{n \text { ones }})+n-1
$$

$$
S_{n}=\frac{1}{9} \sum_{k=1}^{n}\left(10^{k}-1\right)+n-1
$$

$$
S_{n}=\frac{1}{9}\left(10 \cdot \frac{10^{n}-1}{9}-n\right)+n-1=S_{n}=\frac{10^{n+1}+72 n-91}{81}
$$