To create a sequence formula that describes the number of carrots Perry will have on any given day, we first analyze what happens each day.
- On Day 1, Perry starts with 20 carrots: \( a_1 = 20 \).
- On Day 2, he eats half of the carrots, leaving him with half: \( a_2 = 20 - (20 \cdot \frac{1}{2}) = 20 \cdot \frac{1}{2} = 10 \).
- On Day 3, he eats half of what is left (10), so he has: \( a_3 = 10 - (10 \cdot \frac{1}{2}) = 10 \cdot \frac{1}{2} = 5 \).
- On Day 4, he eats half of what's left (5), so he has: \( a_4 = 5 - (5 \cdot \frac{1}{2}) = 5 \cdot \frac{1}{2} = 2.5 \).
- This continues each day, as he eats half of what remains.
From this pattern, we can establish that each day's amount can be expressed as:
\[ a_n = 20 \cdot \left(\frac{1}{2}\right)^{n-1} \]
Thus, the correct expression to describe the number of carrots Perry will have on any given day is:
a_n = 20 \left(\frac{1}{2}\right)^{n-1}
So the correct placement from your available options would be:
- a_n = 20 \left(\frac{1}{2}\right)^{n-1}
This matches our established formula for the number of carrots left each day after he eats half.
The other options do not accurately describe the amount of carrots he has:
-
a_n = 20 - \left(2 * \left(\frac{1}{2}\right)^{n-1}\right): This expression does not properly represent half-eating correctly without adjustments for the alternating days.
-
a_n = \left(\frac{1}{20}\right)^{n-1}: This expression is not relevant to the scenario described.
Therefore, the only appropriate formula for the sequence describing the number of carrots Perry has on day \( n \) is:
- a_n = 20 \left(\frac{1}{2}\right)^{n-1}