Every term of a geometric progression is positive , and every term is the sum of the two preceding terms. Then the common ratio of the geometric progression is:

Step-by-step solution

Let the geometric progression be a, ar, ar^2, Given that every term is positive, we have a > 0 and r > 0 . According to the given condition, every term is the sum of the two preceding terms. T_n = T_ n-1 + T_ n-2 For n = 3 , we get: ar^2 = ar + a Dividing by a (since a > 0 ): r^2 = r + 1 r^2 - r - 1 = 0 Solving for r using the quadratic formula: r = -(-1) (-1)^2 - 4(1)(-1) 2 r = 1 5 2 Since r > 0 , we reject r = 1 - 5 2 . Therefore, the common ratio is r = 1 + 5 2 . Answer: 1 + 5 2