\section*{Problem 1 - 171221}

Determine all real numbers \(a_{1}, d, b_{1}, q\) for which the following statement holds:

If

(1) \(a_{1}\) is the first term and \(d\) is the common difference of an arithmetic sequence \(\left(a_{n}\right)\) and if

(2) \(b_{1}(\neq 0)\) is the first term and \(q\) is the common ratio of a geometric sequence \(\left(b_{n}\right)\), then these sequences have the properties

(3) \(a_{1}=-3 b_{1}\),

(4) \(a_{2}=2 b_{2}\),

(5) \(a_{3}=b_{3}\),

(6) \(d\) is an integer.
}

For the arithmetic sequence, it holds: \(a_{i+1}=a_{i}+d\).

For the geometric sequence, it holds: \(b_{i+1}=b_{i} \cdot q\) with \(q=\frac{b_{i+1}}{b_{i}}\). Using (3), (4), and (5), we can initially determine the equations for \(d\): (i) and for \(q\): (ii).

\[
\begin{gathered}
\frac{b_{2}}{b_{1}}=q=\frac{\frac{a_{2}}{2}}{\frac{a_{1}}{-3}}=-\frac{3}{2} \frac{a_{2}}{a_{1}} \quad ; \quad \frac{b_{3}}{b_{2}}=q=\frac{a_{3}}{\frac{a_{2}}{2}}=2 \frac{a_{3}}{a_{2}} \\
\frac{b_{3}}{b_{2}}=q=\frac{a_{3}}{\frac{a_{2}}{2}}=2 \frac{a_{3}}{a_{2}} \quad ; \quad-\frac{3}{2} \frac{a_{2}}{a_{1}}=\frac{2 a_{3}}{a_{2}} \\
-\frac{3}{2} \frac{a_{2}}{\left(a_{2}-d\right)}=2 \frac{\left(a_{2}+d\right)}{a_{2}} \quad ; \quad-\frac{3}{2} a_{2}^{2}=2\left(a_{2}+d\right)\left(a_{2}-d\right) \\
-\frac{3}{2} a_{2}^{2}=2 a_{2}^{2}-2 d^{2} \quad ; \quad d= \pm \sqrt{7} \cdot \frac{a_{2}}{2} \quad(i) \\
q=2 \frac{a_{3}}{a_{2}}=2 \frac{a_{2}+d}{a_{2}}=a_{2} \frac{(2+\sqrt{7})}{a_{2}}= \pm \sqrt{7}+2 \quad
\end{gathered}
\]

From the sum of the terms \(\sum_{i=1}^{3} a_{i}\), \(b_{2}\) can be calculated, and from (4), \(a_{2}\) can be immediately determined.

\[
\begin{gathered}
a_{1}+a_{2}+a_{3}=a_{2}-d+a_{2}+a_{2}+d=3 a_{2}=-\frac{3 b_{2}}{2+\sqrt{7}}+2 b_{2}+2 b_{2}+\sqrt{7} b_{2} \\
3 a_{2}(2+\sqrt{7})=3 b_{2}+2 b_{2}(2+\sqrt{7})+b_{2} \sqrt{7}(2+\sqrt{7}) \\
6 b_{2}(2+\sqrt{7})=3 b_{2}+4 b_{2}+2 b_{2} \sqrt{7}+2 \sqrt{7} b_{2}+7 b_{2} \\
b_{2}=\sqrt{7} \quad ; \quad a_{2}=2 \sqrt{7}
\end{gathered}
\]

With (i) and (ii), there are two solutions each for \(d= \pm 7\) and \(q= \pm \sqrt{7}+2\), from which the following values are obtained.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline\(d\) & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(q\) & \(b_{1}\) & \(b_{2}\) & \(b_{3}\) & \(a_{1} / b_{1}\) & \(a_{2} / b_{2}\) & \(a_{3} / b_{3}\) \\
\hline 7 & \(2 \sqrt{7}-7\) & \(2 \sqrt{7}\) & \(2 \sqrt{7}+7\) & \(2+\sqrt{7}\) & \(\sqrt{7} /(2+\sqrt{7})\) & \(\sqrt{7}\) & \(\sqrt{7} \cdot(2+\sqrt{7})\) & -3 & 2 & 1 \\
\hline 7 & \(-1.71\) & 5.28 & 12.29 & 4.64 & 0.57 & 2.64 & 12.29 & -3 & 2 & 1 \\
\hline-7 & \(2 \sqrt{7}+7\) & \(2 \sqrt{7}\) & \(2 \sqrt{7}-7\) & \(2-\sqrt{7}\) & \(\sqrt{7} /(2-\sqrt{7})\) & \(\sqrt{7}\) & \(\sqrt{7} \cdot(2-\sqrt{7})\) & -3 & 2 & 1 \\
\hline-7 & 12.29 & 5.28 & \(-1.71\) & \(-0.64\) & \(-4.097\) & 2.64 & \(-1.71\) & -3 & 2 & 1 \\
\hline
\end{tabular}