Given that, the square of the $7^{\text{th}}$ term is equal to the products of the $3^{\text{rd}}$ and $17^{\text{th}}$ terms.
The $n^{\text{th}}$ term of an arithmetic progression $= a + (n-1)d$
Where, $a = $ first term, and $d = $common difference.
Now, ${(a+6d)}^{2} = (a + 2d)(a + 16d)$
$\Rightarrow a^{2} + 36d^{2} + 12ad = a^{2} + 16ad + 2ad + 32d^{2}$
$\Rightarrow 4d^{2} = 6ad$
$\Rightarrow 2d = 3a$
$\Rightarrow 3a = 2d$
$\Rightarrow \frac{a}{d} = \frac{2}{3}$
$\therefore$ The ratio of the first term to the common difference is $2:3.$
Correct Answer $:\text{A}$