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.
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