## Task A-4.1.

Nikola has thought of five numbers. The first number he thought of is -2, and the fifth is 6. The first four numbers are consecutive terms of an arithmetic sequence, and the last three are consecutive terms of a geometric sequence. What numbers did Nikola think of?
## First solution.

Let the common difference of the arithmetic sequence be $d$.

Then the first four numbers are $-2, -2+d, -2+2d, -2+3d$.

Since $-2+2d, -2+3d$, and 6 are consecutive terms of a geometric sequence, we have

$$
\frac{6}{-2+3d}=\frac{-2+3d}{-2+2d}
$$

Rearranging gives

$$
\begin{gathered}
(-2+3d)^{2}=6 \cdot(-2+2d) \\
9d^{2}-24d+16=0
\end{gathered}
$$

from which $d=\frac{4}{3}$.

Nikola thought of the numbers $-2, -\frac{2}{3}, \frac{2}{3}, 2, 6$.