I agree to hire you for 30 days. You can decide between two methods of payments: either (1) $1000 a day, or (2) one penny on the first day, two pennies on the second day, and continue to double your daily pay each day up to day 30. Use quick estimation to make your decision, and justify it.

Number of days you get hired for,$N=30.$

According to the first method of payment, you will get $1000 per day.

According to the second mode of payment, the amount will get doubled each day up to the 30th day.

The first mode is paying $1000 every day for 30 days.

The total money you will earn at the end of 30 days is as follows:

$\begin{array}{c}{A}_1=\$1000\times 30\\ =\$3000\end{array}$

Thus, after 30 days, you will get $3000.

The second mode of payment gives one penny on day 1, two pennies on day 2, and doubles each amount up to the 30th day.

One penny is equal to $0.01.

On the t^th day, the total amount you will get is given by the generalized formula $\$0.01\left(2^{t-1}\right)$.

On the first day (t = 1), you will get $\$0.01\left(2^{1-1}\right)=\$0.01$;

On the second day (t = 2), you will get $\$0.01\left(2^{2-1}\right)=\$0.02$, and so on.

On the last day (t = 30), you will get $\$0.01\left(2^{30-1}\right)=\$5.4\times {10}^6$.

The final amount is over 5 million dollars.

Thus, you should prefer the second method of payment as it gives a much larger amount for 30 days than the first one.