When the same constant is added to the numbers \(60, 100,\) and \(160\) a three-term geometric sequence arises. What is the common ratio of the resulting sequence?
+5 Let the constant be x:
By the definition of a geometric sequence, then: (60 + x)/(100 + x) = (100 + x)/(150+ x)
Cross multiplying: (60 + x)(150 + x) = (100 + x)(100 + x)
---> 9000 + 210x + x² = 10000 + 200x + x²
---> 10x = 1000
---> x = 100
[100 + C] / [60 + C] = [160 + C] / [100 + C], solve for C
Cross Multiply
[100 + C]^2 = [60 + C) * [160 + C]
C^2 + 200 C + 10000 = C^2 + 220 C + 9600
C^2 - C^2 + 200C - 220C ==9600 - 10000
- 20C =- 400
C == - 400 / - 20 = 20
So, the GS would look like this:
[60 + 20], [100 + 20], [160 + 20] = 80, 120, 180
First term = 80 and Common ratio = 120 / 80 = 1.5
