Which would you rather be paid for 12 days - $100 per day for the 12 days or one dollar for the first day, $2.00 the second, $4.00 the third, $8.00 the fourth, etc., up to and including the 12th day?
1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1,024 + 2,048
To determine which option would result in higher pay, let's calculate the amount for each option.
Option 1: $100 per day for 12 days
The total amount for each day is $100, so to calculate the total for 12 days, we multiply $100 by 12:
Total amount = $100/day * 12 days = $1200
Option 2: Doubling amount each day
In this option, the amount for each day doubles compared to the previous day. We can use a formula to calculate the sum of a geometric series to find the total amount.
The formula for the sum of a geometric series is:
Sn = a * (r^n - 1) / (r - 1)
Where:
Sn is the sum of the series,
a is the first term,
r is the common ratio,
n is the number of terms.
In this case:
a = $1
r = 2
n = 12
Plugging the values into the formula:
Total amount = $1 * (2^12 - 1) / (2 - 1)
Total amount = $1 * (4096 - 1)
Total amount = $4095
Comparing the two options:
Option 1: $1200
Option 2: $4095
Given the calculations, the second option would result in a higher pay of $4095 compared to $1200 in the first option. So, it would be preferable to choose the second option where the amount doubles each day.