To illustrate exponential growth where a population of 120 doubles every day, we can use the formula:
\[ P(t) = P_0 \times 2^t \]
Where:
- \( P(t) \) is the population at time \( t \)
- \( P_0 \) is the initial population (120 in this case)
- \( t \) is the number of days that have passed
Let's calculate the population for the first 7 days:
-
Day 0:
- \( P(0) = 120 \times 2^0 = 120 \)
-
Day 1:
- \( P(1) = 120 \times 2^1 = 120 \times 2 = 240 \)
-
Day 2:
- \( P(2) = 120 \times 2^2 = 120 \times 4 = 480 \)
-
Day 3:
- \( P(3) = 120 \times 2^3 = 120 \times 8 = 960 \)
-
Day 4:
- \( P(4) = 120 \times 2^4 = 120 \times 16 = 1920 \)
-
Day 5:
- \( P(5) = 120 \times 2^5 = 120 \times 32 = 3840 \)
-
Day 6:
- \( P(6) = 120 \times 2^6 = 120 \times 64 = 7680 \)
-
Day 7:
- \( P(7) = 120 \times 2^7 = 120 \times 128 = 15360 \)
So, the population at the end of each day would be:
- Day 0: 120
- Day 1: 240
- Day 2: 480
- Day 3: 960
- Day 4: 1920
- Day 5: 3840
- Day 6: 7680
- Day 7: 15360
This sequence demonstrates the rapid increase in population due to exponential growth, with the population doubling every day.
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