Find the geometric progression for which the sum of the first two terms is -4, the fifth term is four times the third term and common ratio is less than zero

Step-by-step solution

Let the first term of the geometric progression be a and the common ratio be r . The terms are a, ar, ar^2, ar^3, ar^4, The sum of the first two terms is a + ar = -4 , which implies a(1 + r) = -4 . (Equation 1) The fifth term is four times the third term, so ar^4 = 4(ar^2) . Since a 0 , we divide by ar^2 to get r^2 = 4 . This gives r = 2 or r = -2 . The problem states that the common ratio is less than zero, so r = -2 . Substituting r = -2 into Equation 1: a(1 - 2) = -4 , which gives -a = -4 , so a = 4 . The terms