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, 32
Joslyn wants to determine the number of daisy seeds, d, she should plant on the nth day. Which statements below are true? Select two that apply.
A.
The sequence is arithmetic.
B.
The sequence is geometric.
C.
The sequence is represented by d=2n.
D.
The sequence is represented by d=2n-1.
E.
The sequence is represented by d=2^n.
F.
The sequence is represented by d=2^n-1^
1, 2, 4, 8, 16, 32
Joslyn wants to determine the number of daisy seeds, d, she should plant on the nth day. Which statements below are true? Select two that apply.
A.
The sequence is arithmetic.
B.
The sequence is geometric.
C.
The sequence is represented by d=2n.
D.
The sequence is represented by d=2n-1.
E.
The sequence is represented by d=2^n.
F.
The sequence is represented by d=2^n-1^
Answers
GPT-4o mini
To determine which statements about the sequence are true, let's analyze it.
The number of seeds planted each day is as follows:
- Day 1: 1
- Day 2: 2
- Day 3: 4
- Day 4: 8
- Day 5: 16
- Day 6: 32
1. **Identify the nature of the sequence:**
- The pattern shows that each day's number of seeds is double the previous day's seeds. For example, 2 is double 1, 4 is double 2, 8 is double 4, etc. This indicates that the sequence is geometric.
2. **Mathematical representation of the sequence:**
- The numbers planted are powers of 2. Specifically,
- Day 1 (n=1): 2^0 = 1
- Day 2 (n=2): 2^1 = 2
- Day 3 (n=3): 2^2 = 4
- Day 4 (n=4): 2^3 = 8
- Day 5 (n=5): 2^4 = 16
- Day 6 (n=6): 2^5 = 32
- This shows that on the nth day, the number of seeds is represented as \( d = 2^{n-1} \).
3. **Evaluate the statements:**
- A. The sequence is arithmetic. **(False)** - An arithmetic sequence has a constant difference; this sequence does not.
- B. The sequence is geometric. **(True)** - Each term is obtained by multiplying the previous term by a constant factor (2).
- C. The sequence is represented by d=2^n. **(False)** - This is incorrect; it should be \( d = 2^{n-1} \).
- D. The sequence is represented by d=2n-1. **(False)** - This does not represent the sequence correctly.
- E. The sequence is represented by d=2^n. **(False)** - As stated, it should be \( d = 2^{n-1} \).
- F. The sequence is represented by d=2^n-1. **(False)** - Also incorrect; it should be \( d = 2^{n-1} \).
The two true statements are:
- **B. The sequence is geometric.**
- (There is an absence of another correct option based on the original choices provided.)
However, if we consider the nature of the sequence aside from the options provided, one could argue about the power representation needed in the mathematical statement regarding day n's terms.
If we modify D to be \( D. \, d \, = \, 2^{n \,- \,1 } \), then this would be a true representation of the sequence as well. But it’s important to note the provided options are indeed limited.
### Summary:
The correct answer is:
- **B. The sequence is geometric.**
- The correct representation isn't accurately listed from the remaining choices.
The number of seeds planted each day is as follows:
- Day 1: 1
- Day 2: 2
- Day 3: 4
- Day 4: 8
- Day 5: 16
- Day 6: 32
1. **Identify the nature of the sequence:**
- The pattern shows that each day's number of seeds is double the previous day's seeds. For example, 2 is double 1, 4 is double 2, 8 is double 4, etc. This indicates that the sequence is geometric.
2. **Mathematical representation of the sequence:**
- The numbers planted are powers of 2. Specifically,
- Day 1 (n=1): 2^0 = 1
- Day 2 (n=2): 2^1 = 2
- Day 3 (n=3): 2^2 = 4
- Day 4 (n=4): 2^3 = 8
- Day 5 (n=5): 2^4 = 16
- Day 6 (n=6): 2^5 = 32
- This shows that on the nth day, the number of seeds is represented as \( d = 2^{n-1} \).
3. **Evaluate the statements:**
- A. The sequence is arithmetic. **(False)** - An arithmetic sequence has a constant difference; this sequence does not.
- B. The sequence is geometric. **(True)** - Each term is obtained by multiplying the previous term by a constant factor (2).
- C. The sequence is represented by d=2^n. **(False)** - This is incorrect; it should be \( d = 2^{n-1} \).
- D. The sequence is represented by d=2n-1. **(False)** - This does not represent the sequence correctly.
- E. The sequence is represented by d=2^n. **(False)** - As stated, it should be \( d = 2^{n-1} \).
- F. The sequence is represented by d=2^n-1. **(False)** - Also incorrect; it should be \( d = 2^{n-1} \).
The two true statements are:
- **B. The sequence is geometric.**
- (There is an absence of another correct option based on the original choices provided.)
However, if we consider the nature of the sequence aside from the options provided, one could argue about the power representation needed in the mathematical statement regarding day n's terms.
If we modify D to be \( D. \, d \, = \, 2^{n \,- \,1 } \), then this would be a true representation of the sequence as well. But it’s important to note the provided options are indeed limited.
### Summary:
The correct answer is:
- **B. The sequence is geometric.**
- The correct representation isn't accurately listed from the remaining choices.