1 cent for the first day, 2 cents for the second day, 4 cents for the third day, 8 cents for the fourth day, etc. If a top of the line Apple Watch costs $800 in

Question

 The Gadget Store offers a one-month Apple Watch rental with payments calculated according to the following formula: 1 cent for the first day, 2 cents for the second day, 4 cents for the third day, 8 cents for the fourth day, etc. If a top of the line Apple Watch costs $800 in the Apple Store, is the one-month rental a good deal? Explain.

Answer 1

The Gadget Store rentals produce the sum of the n digits of a binary number:

  Day 1:       0 0 0 0 1         is 2⁰ = 2¹ - 1 = 1

  Day 2:       0 0 0 1 1         is 2¹ + 2⁰ = 2² - 1 = 3

  Day 3:       0 0 1 1 1         is 2² + 2¹ + 2⁰ = 2³ - 1 = 8 - 1 = 7

  Day 4:       0 1 1 1 1         is 2³ + 2² + 2¹ + 2⁰ = 16 - 1 = 15

  . . .

  Day 15:     0 1 1 1  1 1 1 1  1 1 1 1  1 1 1 1     is 32768 - 1 = $327.67  total so far

  Day 16:     1 1 1 1  1 1 1 1  1 1 1 1  1 1 1 1     is 65536 - 1 = $755,36  total so far

  Day 17:  1 1 1 1 1  1 1 1 1  1 1 1 1  1 1 1 1     is 131072 - 1 = $1310.72 total so far

  . . .

  Day 30 or 31

 

If the watch is rented for more than 16 days, it will cost more than the $800.00 purchase price.

Answer 2

It looks like the charge for n days of rental is 2^n-1¢.  I would look at when 2^n-1 = purchase price = $800 = 80,000¢ and not rent for more than n days.  

 

2^n-1 = 80,000

 

(n-1) log 2 = log 80,000

 

n-1 = (log 80,000) / (log 2) ≅ 16.3

 

Do not rent for more than 17 days, because it is cheaper to just buy it outright.

 

If you meant that the total charge for n days is the sum 2⁰ + 2¹ + 2² + 2³ + ... + 2^n-1, then you need to find the n when the sum just exceeds $800, and rent for no more than 1 day less than that.