You offer to do the dishes for your family for the next month.This month has only 31 days. You suggest that they can pay you in one of three ways:

A. $0.50 each day.(I already did this;=$15.50 a month.
B.$0.10 the first day,$0.20 the second day,$0.30 the third day,and so on.
C.$0.01 1st day,$0.02 2nd day,$0.04 3rd day,doubling each time.

Which rate of pay would be best for you?

Comments
I already know the 1st and 2nd choices,but to calculate the 3rd one one by one seems a bit overrated.
So...I was wondering if there is a sort of pattern for doubling like in the third choice?
Thanks,

Serpentlord123

You get .01 for the first day, .02 for the second, then keep doubling. However, they might fire you after the first dozen days or so.

It is a logarithmic progression in contrast to the second, which is an arithmetic progression.

1 + 2^30 = ?

In currency terms = .01 + .02 + .04 + .08 + .16 + .32 + .64 + 1.28 + 2.56 + 5.12 + 10.24 + 20.48….

To determine which rate of pay would be best for you, let's compare the three options:

Option A: $0.50 each day for 31 days
To find the total earnings for this option, you can simply multiply the daily rate by the number of days:
$0.50 * 31 = $15.50

Option B: Increasing rates each day
In this option, the rate increases gradually each day. To calculate the total earnings for this option, you need to find the sum of an arithmetic series. The formula for the sum of an arithmetic series is given by:
Sum = (n/2) * (first term + last term)

In this case, the first term (a₁) is $0.10, and the last term (aₙ) is $0.10 + (n-1)(0.10). We need to find the value of n, where n represents the number of days.

Using the formula for the last term, we can set it equal to $0.30 (the rate on the third day):
$0.10 + (n-1)(0.10) = $0.30

Simplifying the equation, we get:
0.10n - 0.10 + 0.10 = 0.30
0.10n = 0.30
n = 3

So, the number of days (n) is 3. Now we can calculate the sum of the arithmetic series:
Sum = (n/2) * (first term + last term)
Sum = (3/2) * ($0.10 + $0.30)
Sum = 1.5 * $0.40
Sum = $0.60

Therefore, the total earnings for Option B would be $0.60.

Option C: Doubling rates each day
In this option, the rate doubles each day. To calculate the total earnings for this option, we can use the formula for the sum of a geometric series. The formula is given by:
Sum = (a₁ * (rⁿ - 1)) / (r - 1)

In this case, the first term (a₁) is $0.01, and the common ratio (r) is 2. We need to calculate the value when n equals 31 since there are 31 days in the month.

Using the formula for the sum of the geometric series, we can find the total earnings for Option C:
Sum = ($0.01 * (2³¹ - 1)) / (2 - 1)
Sum = ($0.01 * (2³¹ - 1)) / 1
Sum = ($0.01 * (2³¹ - 1))
Sum ≈ $21.47

Therefore, the total earnings for Option C would be approximately $21.47.

Comparing the three options, we find that:
Option A gives you $15.50
Option B gives you $0.60
Option C gives you approximately $21.47

Hence, the best rate of pay for you would be Option C, where you earn approximately $21.47 in a month.