Let a₁, a₂, a₃, be a geometric progression of positive integers such that a₁ = 3 and a_ n+2 - 2a_n = a_ n+1 for all positive integers n . What is the value of a₁ + a₂ + a₃ + a₄ + a₅ ?

Step-by-step solution

Let the common ratio of the geometric progression be r . The n -th term is given by a_n = a₁ r^ n-1 . Given the recurrence relation a_ n+2 - a_ n+1 - 2a_n = 0 . Substituting a_n = a₁ r^ n-1 , we get a₁ r^ n+1 - a₁ r^n - 2a₁ r^ n-1 = 0 . Dividing by a₁ r^ n-1 (since a₁ = 3 and r 0 ), we obtain the characteristic equation: r^2 - r - 2 = 0 Factoring the quadratic equation yields (r - 2)(r + 1) = 0 . Thus, r = 2 or r = -1 . Since the sequence consists of positive integers, the common ratio r must be positive. Therefore