Perry buys a bag of 20 carrots on the first day. The second day, he eats half of the bag. Each day after, he eats half of what is left in the bag. Create a sequence formula that describes the number of carrots Perry will have on any given day.

Question

Drag and drop the expressions into the correct locations:

- a_n = 20 \left((1)/(2)\right)^(n-1)
- a_n = 20 - \left(2 * \left((1)/(2)\right)^(n-1)\right)
- a_n = \left((1)/(20)\right)^(n-1)

Answer 1

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}