Theory question 101
Progressions(a) Given that (7 − 2x), 9, (5x + 17) are consecutive terms of a Geometric Progression (G.P.) with common ratio, r > 0, find the values of x.
(b) Two positive numbers are in the ratio 3 : 4. The sum of thrice the first number and twice the second is 68. Find the smaller number.
Show the model answer
Model answer
(a) Middle term squared = product of neighbours: 81 = (7 − 2x)(5x + 17)
81 = 35x + 119 − 10x2 − 34x → 10x2 − x − 38 = 0
(10x + 19)(x − 2) = 0 → x = 2 or x = −19/10
Check r > 0: x = 2 gives 3, 9, 27 (r = 3); x = −1.9 gives 10.8, 9, 7.5 (r = 5/6). Both are valid
(b) Numbers 3k and 4k: 9k + 8k = 68 → k = 4
Smaller number = 12
Explanation
In a G.P., T2/T1 = T3/T2.