To model a mice population that starts at 1 and doubles every year, you can use an exponential growth formula. The general formula for exponential growth is:
\[ P(t) = P_0 \cdot r^t \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (in this case, since the population doubles, \( r = 2 \)),
- \( t \) is the time in years.
For your specific scenario, where the initial population \( P_0 = 1 \) and the population doubles every year, the formula becomes:
\[ P(t) = 1 \cdot 2^t \]
or simply:
\[ P(t) = 2^t \]
This formula indicates that the population of mice at any time \( t \) in years is given by \( 2^t \). For example:
- At \( t = 0 \) (start), \( P(0) = 2^0 = 1 \) mouse.
- At \( t = 1 \) (after 1 year), \( P(1) = 2^1 = 2 \) mice.
- At \( t = 2 \) (after 2 years), \( P(2) = 2^2 = 4 \) mice.
- At \( t = 3 \) (after 3 years), \( P(3) = 2^3 = 8 \) mice, and so on.
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