## Task 4 - 211234

Determine all non-zero real numbers $q$ that have the following property: There exists a non-zero number $a_{1}$ and a natural number $k \geq 3$ such that in the sequence $\left(a_{n}\right)$ defined by $a_{n}=a_{1} \cdot q^{n-1}(n=1,2,3, \ldots)$, the term $a_{k}$ is equal to the arithmetic mean of the two preceding terms $a_{k-1}$ and $a_{k-2}$.
By assumption, there exist $a_{1}, k, q$, such that

$$
a_{1} q^{k-1}=a_{k}=\frac{1}{2}\left(a_{k-1}+a_{k-2}\right)=\frac{1}{2}\left(a_{1} q^{k-2}+a_{1} q^{k-3}\right)
$$

Cancelling with $a_{1}, q^{k-3}$ yields $q^{2}=\frac{1}{2}(q+1)$ and thus $q=\frac{1}{2}(1 \pm \sqrt{3})$.

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