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?

Concept:

In an AP, the nth term is given by:

T_n = a + (n - 1)d

In a GP, terms satisfy the relation:

T_m² = T_n × T_p

Calculation:

Let the 5th, 7th, and 13th terms of the AP be T₅, T₇, and T₁₃ respectively.

⇒ T₅ = a + 4d

⇒ T₇ = a + 6d

⇒ T₁₃ = a + 12d

Since the terms are in GP:

⇒ T₇² = T₅ × T₁₃

⇒ (a + 6d)² = (a + 4d) × (a + 12d)

Expanding both sides:

⇒ (a + 6d)² = a² + 12ad + 48d²

⇒ a² + 12ad + 36d² = a² + 12ad + 48d²

Cancel out common terms:

⇒ 36d² = 48d²

⇒ -12d² = 0

Dividing through by d² (assuming d ≠ 0):

⇒ a = -3d

Conclusion:

∴ Ratio of the first term to the common difference is:

⇒ a/d = -3

Hence, the correct answer is Option 1.