Let t₁, t₂, t₃, be a strictly increasing geometric progression of positive real numbers. If the product of the second, third, and fourth terms is 1728 , and the sum of the fourth and fifth terms is 72 , then the value of t₁ + t₂ is equal to

Step-by-step solution

Given the product of the second, third, and fourth terms is 1728 : t₂ t₃ t₄ = 1728 Using the property of a geometric progression, t₂ t₄ = t₃^2 . Substituting this into the product gives: t₃^3 = 1728 t₃ = 12 We are also given that the sum of the fourth and fifth terms is 72 : t₄ + t₅ = 72 Expressing t₄ and t₅ in terms of t₃ and the common ratio r : t₃ r + t₃ r^2 = 72 12r + 12r^2 = 72 r^2 + r - 6 = 0 (r + 3)(r - 2) = 0 This yields r = -3 or r = 2 . Since the geometric progression is strictly increasing and consists o