The terms of an infinitely decreasing geometric progression in which all the terms are positive, the first term is 4 , and the difference between third and fifth term is 32 81 , then which of the following is not true

Step-by-step solution

Let the first term be a = 4 and the common ratio be r , where 0 The third term is T₃ = ar^2 = 4r^2 and the fifth term is T₅ = ar^4 = 4r^4 . Given T₃ - T₅ = 32 81 , we have 4r^2 - 4r^4 = 32 81 . Dividing by 4, r^2 - r^4 = 8 81 , which implies r^4 - r^2 + 8 81 = 0 . Let x = r^2 . Then x^2 - x + 8 81 = 0 . Solving for x using the quadratic formula x = -b b^2 - 4ac 2a : x = 1 1 - 4(8/81) 2 = 1 1 - 32/81 2 = 1 49/81 2 = 1 7/9 2 . The two possible values for x = r^2 are x₁ = 1 + 7/9 2 = 16/9 2 = 8 9 and x₂ = 1 - 7/9 2 =