41. Find all geometric progressions in which each term, starting from the third, is equal to the sum of the two preceding ones.
41. Let us consider such a geometric progression. Then for it, the equality holds

$$
u_{1} q^{n+1}=u_{1} q^{n}+u_{1} q^{n-1} \text { for all } n \text {. }
$$

From this, it follows that

$$
\begin{gathered}
q^{2}=q+1 \\
q=\frac{1 \pm \sqrt{5}}{2}
\end{gathered}
$$

Thus, the common ratio of a geometric progression, in which each term is equal to the sum of the two preceding terms, can only take the values $\frac{1+\sqrt{5}}{2}$ and $\frac{1-\sqrt{5}}{2}$. It is clear that the converse is also true, i.e., that in any progression with one of these common ratios, each term is equal to the sum of the two preceding terms.