If 5th, 7th and 13th terms of an AP are in GP, then what is the...

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: $-3$