Theory question 106
ProgressionsGiven that (x + 2), (4x + 3) and (7x + 24) are consecutive terms of a Geometric Progression (G.P.), find the: (a) value of x; (b) common ratio.
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Model answer
(a) (4x + 3)/(x + 2) = (7x + 24)/(4x + 3) ⇒ (4x + 3)² = (x + 2)(7x + 24)
16x² + 24x + 9 = 7x² + 38x + 48 ⇒ 9x² − 14x − 39 = 0 ⇒ (x − 3)(9x + 13) = 0
x = 3 (or x = −13/9)
(b) With x = 3 the terms are 5, 15, 45, so the common ratio = 3 (with x = −13/9 the terms are 5/9, −25/9, 125/9 and the ratio is −5)
Explanation
In a G.P. the middle term squared equals the product of its neighbours.