Task B-4.2. Three numbers form an increasing arithmetic sequence. If the second number is increased by 1, and the third by 10, the sequence becomes geometric. The smallest of these three numbers is 2. What are these numbers?
## Solution.

If the smallest term of an increasing arithmetic sequence is equal to 2, then the given numbers are of the form

$$
2, \quad 2+d, \quad 2+2 d
$$

Furthermore, the geometric sequence consists of the numbers

$$
2, \quad 3+d, \quad 12+2 d
$$

so it follows that $(3+d)^{2}=2 \cdot(12+2 d)$.

The solutions to this equation are $d_{1}=3, d_{2}=-5$.

Since the sequence is increasing, the only solution is $d=3$, and the desired numbers are $2,5,8$.

(1 point)