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# geometric progression

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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?

Sep 8, 2023

#1
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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

.

Sep 8, 2023

#1
+1

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