If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference?

Let the first term of the AP be $a$ and the common difference be $d$.

The 5th, 7th, and 13th terms of the AP are $a + 4d$, $a + 6d$, and $a + 12d$ respectively.

Since these terms are in GP, the square of the middle term is equal to the product of the other two terms:

$(a + 6d)^2 = (a + 4d)(a + 12d)$

Expanding both sides:

$a^2 + 12ad + 36d^2 = a^2 + 16ad + 48d^2$

Simplifying the equation:

$4ad + 12d^2 = 0$

$4d(a + 3d) = 0$

Assuming $d \neq 0$ for a non-constant AP, we get:

$a + 3d = 0$

$a = -3d$

The ratio of the first term to the common difference is $\dfrac{a}{d} = -3$.

Answer: $-3$