Theory question 113
Progressions(a) If 3, x, y, 18 are in Arithmetic Progression (A.P), find the values of x and y.
(b)(i) The sum of the second and third terms of a geometric progression is six times the fourth term. Find the two possible values of the common ratio. (ii) If the second term is 8 and the common ratio is positive, find the first six terms.
Show the model answer
Model answer
(a) 18 = 3 + 3d → d = 5, so x = 8 and y = 13.
(b)(i) ar + ar2 = 6ar3 → 6r2 − r − 1 = 0 → (3r + 1)(2r − 1) = 0 → r = ½ or r = −⅓.
(ii) r = ½ and ar = 8, so a = 16. First six terms: 16, 8, 4, 2, 1, ½.
Explanation
Divide the GP equation by ar (not zero) to get a quadratic in r.