Write an equation for option 2 and find how much money would you make on the 30th day.

Question

Suppose a retired professor who lives next door offers to pay you to work for him for the next 30 days. He says he will pay you $500 per day (option 1), or if you would like, he will pay you $0.01 for the first day and double your pay each day after that (option 2).

Answer 1

Option 2 follows a Geometric sequence, with the first term = $0.01 and the common ratio = 2.

 

To find the common ratio, divide the proceeding term by the previous term.

 

For example, for this sequence 1st 4 terms will be: 0.01, 0.02, 0.04, 0.08,...

 

Common ratio = 0.02/0.01 or 0.04/0.02, or 0.08/0.04. All these ratios will be equal to 2.

 

Let the 1st term = a, common ratio = r, and the number of terms = n.

 

Therefore, the nth term can be represented by: a_n = ar^n-1

 

Equation for option 2 is : a_n = 0.01(2)^n-1

 

To find the amount made on the 30th day, we set:

 

a = 0.01, r = 2, and n = 30

 

a₃₀ = 0.01(2)^30-1 = 0.01(2)²⁹ = 0.01(536870912)

 

a₃₀ = 5368709.12

 

Therefore, on the 30th day, you will make $5,368,709.12

 

The amount looks big, but you can check manually to see that the answer is correct.

Answer 2

($500 dollars/day)*(30 days) = $15000       [Option 1 may sound good, but is it better?]

 

Option 2:

  $0.01 + $0.02 + $0.04 + $0.08 + . . . + ($0.01)*2^(30-1)

 

  $0.01 + $0.02 + $0.04 + $0.08 + . . . + ($0.01)*2²⁹

  $0.01 + $0.02 + $0.04 + $0.08 + . . . + $5,368,709.12

 

On the thirtieth day, you would be paid $5,368,709.12.         ($0.01)*2^n-1, where n is the day