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By adding the same constant to each of the numbers 20, 50, 100, we get a geometric sequence.  Find the common ratio of the geometric sequence.

 May 19, 2020
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We need to add the same amount to 20, 50, and 100 to get a geometric sequence.

 

Call this amount 'x".

Then our terms will be   t1  =  20 +x,   t2  =  50 + x,   and   t3  =  100 + x

 

To be a geometric sequence, the terms  t1,  t2,  and  t3  must fit in this proportion:

    t3 / t2  =  t2 / t1

 

Substitutng:            (100 + x) / (50 + x)  =  (50 + x) / (20 + x)

Cross-multiplying:  (100 + x) · (20 + x)  =  (50 + x) · (50 + x)

Multiplying out:         2000 + 120x + x2  =  2500 + 100x + x2 

Simplifying:                                    20x  =  500

                                                          x  =  25

 

So, the numbers are  45,  75,  and  125

 

The ratio:  125 / 75  =  5/3

and              75 / 45  =  5/3

 May 19, 2020

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