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?

Step-by-step solution

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 0 for a non-constant AP, we get: a + 3d = 0 a = -3d The ratio of the first term to the common diff