This question will help you appreciate the power of exponential growth (i.e., similar to how rapidly the universe grew during inflation). On your first day of work at a new job, your boss offers you a choice of salaries. Option A is a fixed salary of $5000 per month. Or Option B, which is you can be paid on a daily basis throughout the month, starting with only 1 cent the first day, 2 cents the second day, 4 cents on the third day, etc. Assuming that a month has 21 workdays, which option should you take? (Hint: you’ll get paid 2(n-1) cents on the nth day.)

Question

This question will help you appreciate the power of exponential growth (i.e., similar to how rapidly the universe grew during inflation). On your first day of work at a new job, your boss offers you a choice of salaries. Option A is a fixed salary of $5000 per month. Or Option B, which is you can be paid on a daily basis throughout the month, starting with only 1 cent the first day, 2 cents the second day, 4 cents on the third day, etc. Assuming that a month has 21 workdays, which option should you take? (Hint: you’ll get paid 2(n-1) cents on the nth day.)

BlackTom AI · Accepted Answer

First, restate the question and options to ensure clarity: The question compares two salary structures over a 21-workday month. Option A pays a fixed $5000 per month. Option B pays daily with exponential growth, starting at 1 cent on day 1, then 2 cents on day 2, 4 cents on day 3, and so on, with the hint that on the nth day you receive 2^(n-1) cents.

Analysis of Option A: The fixed salary provides a known total of $5000 for the month, regardless of how the days unfold. This amount is straightforward and does not depend on the number of workdays beyond the assumption of 21 workdays. It is simple, predictable, and does not involve growth over time.

Analysis of Option B: The daily payment forms a geometric progression in cents: 1, 2, 4, 8, ..., up to day 21. The sum of a geometric series with first term a = 1 cent and ratio r = 2 is S_n = a*(r^n - 1)/(r - 1). Here n = 21, so S_21 = 1*(2^21 - 1)/(2 - 1) = 2^21 - 1 = 2,097,151 cents.

Converting to dollars, 2,097,151 cents equals $20,971.51. This total is substantially larger than the fixed $5000 per month.

Why Option B exceeds Option A: The exponential growth doubles the amount each day, so the latter days contribute increasingly large sums, dominating the total. Even though the first days pay very little, the cumulative effect over 21 days yields a sum in the tens of thousands of dollars, far surpassing the fixed monthly salary.

Why the other interpretation might be tempting to some: It might seem that the average daily amount is around half of the final day's value, or that the total could be modest due to early small payments. However, for a doubling sequence, later days contribute enormous amounts, making the total much larger than a flat monthly rate.

Putting it all together, the option with exponential daily pay rapidly accumulates to a much larger total than a fixed $5000 per month, due to the properties of geometric growth in cents and the 21-day horizon.

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