To describe the number of carrots Perry has on any given day, we can define a sequence based on his eating pattern.
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On Day 1 (the first day), he starts with 20 carrots. \[ a_1 = 20 \]
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On Day 2, he eats half of the carrots, so he has: \[ a_2 = \frac{1}{2} a_1 = \frac{1}{2} \times 20 = 10 \]
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On Day 3, he eats half of what is left from Day 2, which is: \[ a_3 = \frac{1}{2} a_2 = \frac{1}{2} \times 10 = 5 \]
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On Day 4, he continues this pattern: \[ a_4 = \frac{1}{2} a_3 = \frac{1}{2} \times 5 = 2.5 \]
Observing the pattern, we can see that each day he has half of what he had the day before. Therefore, the sequence can be generalized using the formula for the number of carrots Perry has on day \( n \):
\[ a_n = \frac{1}{2} a_{n-1} \]
This can be expressed more explicitly as:
\[ a_n = \frac{20}{2^{n-1}} \]
Where:
- \( a_n \) is the number of carrots on day \( n \),
- \( 20 \) is the initial number of carrots,
- \( 2^{n-1} \) represents the number of times he has halved the amount from the initial amount.
Thus, the formula for the number of carrots Perry will have on any given day \( n \) is:
\[ a_n = \frac{20}{2^{n-1}} \]