Arithmetic sequence to geometric sequence. 
The numbers $a_1, a_2, a_3, . . .$ form an arithmetic sequence with $a_1 \ne a_2$. 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.

This is from Euclid 2015:Problem 7B
I got that:
$a_n = a_1 + (n-1)d = r^{n-1} a_1$ where $d$ is difference, and $r$ the geom. ratio.
But this doesnt give many leads, HINTS ONLY PLEASE.

HINT : 
Use the fact that
$$\text{$a,b,c$ form a geometric sequence}\Rightarrow b^2=ac$$
Thus, you have
$$a_2^2=a_1a_6\quad\text{and}\quad a_4^2=a_1a_k$$
Express these by $a_1$ and $d$.
