Gifford earns money by shoveling for the winter. He offers two payment plans: either pay $400 per week for the entire winter or pay $5 for the first week, $10 for the second week, $20 for the third week, and so on. Explain why each plan does or does not form a geometric sequence. Then determine the number of weeks after which the total cost of the second plan will exceed the total cost of the first plan.

Plan 1 is arithmetic and Plan 2 is geometric. Week 10 is when plan 2 exceeds the cost of plan 1.

Step-by-step explanation:

The first payment plan offers $400 each week.

1 400

2 800

3 1200

4 1600

...

This plan is arithmetic since you add 400 for each week. Its represented by the equation y = 400x.

The second plan is $5 the first week and then doubles each week.

1 5

2 10

3 20

4 40

...

This plan is geometric since each week is multiplied by 2 for the next. It's represented by the equation y = 5 (2^x).

At week 10 plan 2 becomes more expensive.

Plan 1 - y = 400 (10) = 4000

Plan 2 - y = 5 (2^10) = 5120