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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?

 
 Nov 24, 2022
 #1
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Using an averaging technique, I found that the answer is 40.

 
 Nov 24, 2022
 #3
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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 

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 Nov 24, 2022
 #4
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[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

 
 Nov 24, 2022

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