The term of the given G.P. are (x + 2), (4x + 3), and (7x + 24) (a) Common ratio = \(\frac{(4x + 3)}{(x + 2)}\) = \(\frac{(7x + 24)}{(4x + 3)}\) cross multiply (4x + 3)(4x + 3) = (x + 2)(7x + 24) 16x\(^2\) + 24x + 9 = 7x\(^2\) + 38x + 48 16x\(^2\) - 7x\(^2\) + 24x - 38x + 9 - 48 = 0 9x\(^2\) - 14x - 39 = 0 9x\(^2\) - 27x + 13x - 39 = 0 (from 9x\(^2\) x 39) 9x(x - 3) + 13(x -3) = 0 (9x + 13)(x - 3) = 0 (9x + 13) = 0 or (x - 3) = 0 x = \(\frac{-13}{9}\) or 3 (b) When x = \(\frac{-13}{9}\) Common ratio = \(\frac{(4x + 3)}{(x + 2)}\) = \(\frac{(4(\frac{-13}{9}) + 3)}{((\frac{-13}{9}) + 2)}\) = \(\frac{\frac{-25}{9}}{\frac{5}{9}}\) = \(\frac{- 25}{9}\) x \(\frac{9}{5}\) = - 5. Common ratio,r = - 5 When x = 3, Common ratio = \(\frac{(4(3) + 3)}{((3) + 2)}\) = \(\frac{15}{5}\) = 3. Common ratio, r = 3