Question
Joslyn is planting daisy seeds in a flower garden. Each day, she plants a different number of seeds. The list below shows the number of seeds Joslyn planted on each of the first six days. 1 , 2 , 4 , 8 , 16 , 321, 2, 4, 8, 16, 32 Joslyn wants to determine the number of daisy seeds, "d"d, she should plant on the "n" "t" "h"nth day. Which statements below are true? Select TWO that apply. (Select all that apply.) "d" equals 2 "n" "d" equals 2 "n" minus 1 "d" equals 2 Superscript, "n" , Baseline "d" equals 2 Superscript, "n" minus 1 , Baseline . When appropriate, include why the answer you choose is correct and why the other choices are not correct.
Answers
GPT-4o mini
To analyze the sequence of seeds Joslyn is planting, we first need to examine the numbers she planted over the first six days:
- Day 1: 1 seed
- Day 2: 2 seeds
- Day 3: 4 seeds
- Day 4: 8 seeds
- Day 5: 16 seeds
- Day 6: 32 seeds
From this sequence, we can observe that the number of seeds planted each day follows a pattern of powers of 2:
- Day 1: \(2^0 = 1\)
- Day 2: \(2^1 = 2\)
- Day 3: \(2^2 = 4\)
- Day 4: \(2^3 = 8\)
- Day 5: \(2^4 = 16\)
- Day 6: \(2^5 = 32\)
This suggests that the general formula for the number of seeds "d" she should plant on the "n"th day can be expressed as:
\[ d = 2^{(n-1)} \]
Now let's investigate the options provided:
1. **"d" equals 2 "n"** - This is incorrect. This does not match our formula, and it does not accurately describe the pattern observed.
2. **"d" equals 2 "n" minus 1** - This is likely typed incorrectly but might imply \(d = 2n - 1\). However, this equation does not fit the observed pattern.
3. **"d" equals 2 Superscript, "n"** - This is also incorrect, as this would suggest \(d = 2^n\), which does not match the pattern since the actual relationship is through \(2^{(n-1)}\).
4. **"d" equals 2 Superscript, "n" minus 1** - This is correct and matches our formal expression of \(d = 2^{(n-1)}\).
Thus, the two correct statements in the context of the sequence and formula are:
- **"d" equals 2 Superscript, "n" minus 1** (True)
- Additionally, none of the other statements are true.
In conclusion, the true statement from the given options is the last one, as it accurately reflects the growth of daisy seeds Joslyn plants each day according to the observed doubling pattern.
- Day 1: 1 seed
- Day 2: 2 seeds
- Day 3: 4 seeds
- Day 4: 8 seeds
- Day 5: 16 seeds
- Day 6: 32 seeds
From this sequence, we can observe that the number of seeds planted each day follows a pattern of powers of 2:
- Day 1: \(2^0 = 1\)
- Day 2: \(2^1 = 2\)
- Day 3: \(2^2 = 4\)
- Day 4: \(2^3 = 8\)
- Day 5: \(2^4 = 16\)
- Day 6: \(2^5 = 32\)
This suggests that the general formula for the number of seeds "d" she should plant on the "n"th day can be expressed as:
\[ d = 2^{(n-1)} \]
Now let's investigate the options provided:
1. **"d" equals 2 "n"** - This is incorrect. This does not match our formula, and it does not accurately describe the pattern observed.
2. **"d" equals 2 "n" minus 1** - This is likely typed incorrectly but might imply \(d = 2n - 1\). However, this equation does not fit the observed pattern.
3. **"d" equals 2 Superscript, "n"** - This is also incorrect, as this would suggest \(d = 2^n\), which does not match the pattern since the actual relationship is through \(2^{(n-1)}\).
4. **"d" equals 2 Superscript, "n" minus 1** - This is correct and matches our formal expression of \(d = 2^{(n-1)}\).
Thus, the two correct statements in the context of the sequence and formula are:
- **"d" equals 2 Superscript, "n" minus 1** (True)
- Additionally, none of the other statements are true.
In conclusion, the true statement from the given options is the last one, as it accurately reflects the growth of daisy seeds Joslyn plants each day according to the observed doubling pattern.