Question

People often underestimate how quickly repeated doubling grows. Try answering this question with a quick guess: If you earn $1 today, $2 tomorrow, $4 the next day, etc., about how much will you earn in a year?

Answer 1

If you earn $1 today, $2 tomorrow, $4 the next day, and continue doubling each day, the amount you earn on day \( n \) can be represented as \( 2^{(n-1)} \).

To find out how much you'll earn in a year (365 days), you can calculate the earnings on the last day:

\[
\text{Earnings on day 365} = 2^{364}
\]

Calculating \( 2^{364} \) gives an extremely large number. In fact, \( 2^{10} = 1024 \) (which is roughly a thousand), and \( 2^{20} = 1,048,576 \) (over a million).

Continuing this exponential growth,
- \( 2^{30} \) is about 1 billion,
- \( 2^{40} \) is about 1 trillion,
- \( 2^{50} \) is about 1 quadrillion.

At \( n = 364 \):

\[
2^{364} \approx 2^{10 \times 36.4} \approx 10^{110.4}
\]

This is roughly \( 1.25 \times 10^{110} \), which is an incomprehensibly large number!

Thus, the amount you would earn in a year with this doubling strategy would be astronomically large, specifically \( 2^{364} \), which is far beyond typical financial or physical quantities.