geometric progression

 

When the same constant is added to the numbers $60,$ $100,$ and $110,$ a three-term geometric sequence arises. What is the common ratio of the resulting sequence?   

 

In a geometric progression the ratio between succeeding numbers is the same.   

 

Therefore                                            100 + x         110 + x    

                                                            ––––––   =   ––––––     

                                                             60 + x          100 + x     

 

Cross multiplying                                 (x + 100)(x + 100)  =  (x + 60)(x + 110)    

                                                             x² + 200x +10,000  =  x² + 170x + 6600       

 

Subtract x² from both sides                  200x + 10,000  =  170x + 6600    

Subtract 170x from both sides                30x + 10,000  =  6600   

Subtract 10,000 from both sides                            30x  =  –3400       

 

                                                                     x  =  –113.333 • • •   

 

Plugging –113.333   

into the first ratio                                       –13.333   

                                                                 –––––––  =  0.24999  =  0.25   

                                                                  –53.333    

Plugging –113.333  

into the second ratio                                  –3.333   

                                                                 –––––––  =  0.24998  =  0.25    

                                                                  –13.333               

Who'd a thunk it would really work.  

 

The common ratio, after     

adding the constant, is                               0.25    

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