Problem (Algebra):
The numbers $a_{1}, a_{2}, a_{3}, \ldots$ form an arithmetic sequence with $a_{1} \neq a_{2}$. The three numbers $a_{1}, a_{2}, a_{6}$ form a geometric sequence in that order. Determine all possible positive integers $k$ for which the three numbers $a_{1}, a_{4}, a_{k}$ also form a geometric sequence in that order.

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 are the first four terms of an arithmetic sequence.

A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant. For example, $3,6,12$ is a geometric sequence with three terms.)

Solution:
Suppose that the arithmetic sequence $a_{1}, a_{2}, a_{3}, \ldots$ has first term $a$ and common difference $d$.

Then, for each positive integer $n, a_{n}=a+(n-1) d$.

Since $a_{1}=a$ and $a_{2}=a+d$ and $a_{1} \neq a_{2}$, then $d \neq 0$.

Since $a_{1}, a_{2}, a_{6}$ form a geometric sequence in that order, then $\frac{a_{2}}{a_{1}}=\frac{a_{6}}{a_{2}}$ or $\left(a_{2}\right)^{2}=a_{1} a_{6}$.

Substituting, we obtain

$$
\begin{aligned}
(a+d)^{2} & =a(a+5 d) \\
a^{2}+2 a d+d^{2} & =a^{2}+5 a d \\
d^{2} & =3 a d \\
d & =3 a \quad(\text { since } d \neq 0)
\end{aligned}
$$

Therefore, $a_{n}=a+(n-1) d=a+(n-1)(3 a)=(3 n-2) a$ for each $n \geq 1$.

Thus, $a_{4}=(3(4)-2) a=10 a$, and $a_{k}=(3 k-2) a$. (Note that $a_{1}=(3(1)-2) a=a$.)

For $a_{1}, a_{4}, a_{k}$ to also form a geometric sequence then, as above, $\left(a_{4}\right)^{2}=a_{1} a_{k}$, and so

$$
\begin{aligned}
(10 a)^{2} & =(a)((3 k-2) a) \\
100 a^{2} & =(3 k-2) a^{2}
\end{aligned}
$$

Since $d \neq 0$ and $d=3 a$, then $a \neq 0$.

Since $100 a^{2}=(3 k-2) a^{2}$ and $a \neq 0$, then $100=3 k-2$ and so $3 k=102$ or $k=34$.

Checking, we note that $a_{1}=a, a_{4}=10 a$ and $a_{34}=100 a$ which form a geometric sequence with common ratio 10 .

Therefore, the only possible value of $k$ is $k=34$.

Final answer: 34