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=n^2
.
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=n^2
.
F.
The sequence is represented by d=2^n-1
.
Answers
GPT-4o mini
To analyze the sequence of daisy seeds planted by Joslyn over the first six days (1, 2, 4, 8, 16, 32), let's evaluate each option:
1. **A. The sequence is arithmetic.**
- An arithmetic sequence has a constant difference between consecutive terms. The differences here are:
2 - 1 = 1
4 - 2 = 2
8 - 4 = 4
16 - 8 = 8
32 - 16 = 16
- Since the differences are not constant, this statement is **false.**
2. **B. The sequence is geometric.**
- A geometric sequence has a constant ratio between consecutive terms. The ratios here are:
2 / 1 = 2
4 / 2 = 2
8 / 4 = 2
16 / 8 = 2
32 / 16 = 2
- Since the ratios are constant (each term is multiplied by 2), this statement is **true.**
3. **C. The sequence is represented by d = 2^n.**
- Evaluating this, we see:
For n = 1, d = 2^1 = 2
For n = 2, d = 2^2 = 4
For n = 3, d = 2^3 = 8
For n = 4, d = 2^4 = 16
For n = 5, d = 2^5 = 32
For n = 6, d = 2^6 = 64 (not part of the sequence)
- This does not match the values given, so this statement is **false.**
4. **D. The sequence is represented by d = 2^(n-1).**
- Evaluating this, we see:
For n = 1, d = 2^(1-1) = 2^0 = 1
For n = 2, d = 2^(2-1) = 2^1 = 2
For n = 3, d = 2^(3-1) = 2^2 = 4
For n = 4, d = 2^(4-1) = 2^3 = 8
For n = 5, d = 2^(5-1) = 2^4 = 16
For n = 6, d = 2^(6-1) = 2^5 = 32
- This matches perfectly with the sequence, so this statement is **true.**
5. **E. The sequence is represented by d = n^2.**
- Evaluating this, we see:
For n = 1, d = 1^2 = 1
For n = 2, d = 2^2 = 4
For n = 3, d = 3^2 = 9 (not part of the sequence)
- This does not match the values given, so this statement is **false.**
6. **F. The sequence is represented by d = 2^n - 1.**
- Evaluating this, we see:
For n = 1, d = 2^1 - 1 = 1
For n = 2, d = 2^2 - 1 = 3 (not part of the sequence)
For n = 3, d = 2^3 - 1 = 7 (not part of the sequence)
- This does not match the values given, so this statement is **false.**
Based on this evaluation, the two true statements are:
**B. The sequence is geometric.**
**D. The sequence is represented by d = 2^(n-1).**
1. **A. The sequence is arithmetic.**
- An arithmetic sequence has a constant difference between consecutive terms. The differences here are:
2 - 1 = 1
4 - 2 = 2
8 - 4 = 4
16 - 8 = 8
32 - 16 = 16
- Since the differences are not constant, this statement is **false.**
2. **B. The sequence is geometric.**
- A geometric sequence has a constant ratio between consecutive terms. The ratios here are:
2 / 1 = 2
4 / 2 = 2
8 / 4 = 2
16 / 8 = 2
32 / 16 = 2
- Since the ratios are constant (each term is multiplied by 2), this statement is **true.**
3. **C. The sequence is represented by d = 2^n.**
- Evaluating this, we see:
For n = 1, d = 2^1 = 2
For n = 2, d = 2^2 = 4
For n = 3, d = 2^3 = 8
For n = 4, d = 2^4 = 16
For n = 5, d = 2^5 = 32
For n = 6, d = 2^6 = 64 (not part of the sequence)
- This does not match the values given, so this statement is **false.**
4. **D. The sequence is represented by d = 2^(n-1).**
- Evaluating this, we see:
For n = 1, d = 2^(1-1) = 2^0 = 1
For n = 2, d = 2^(2-1) = 2^1 = 2
For n = 3, d = 2^(3-1) = 2^2 = 4
For n = 4, d = 2^(4-1) = 2^3 = 8
For n = 5, d = 2^(5-1) = 2^4 = 16
For n = 6, d = 2^(6-1) = 2^5 = 32
- This matches perfectly with the sequence, so this statement is **true.**
5. **E. The sequence is represented by d = n^2.**
- Evaluating this, we see:
For n = 1, d = 1^2 = 1
For n = 2, d = 2^2 = 4
For n = 3, d = 3^2 = 9 (not part of the sequence)
- This does not match the values given, so this statement is **false.**
6. **F. The sequence is represented by d = 2^n - 1.**
- Evaluating this, we see:
For n = 1, d = 2^1 - 1 = 1
For n = 2, d = 2^2 - 1 = 3 (not part of the sequence)
For n = 3, d = 2^3 - 1 = 7 (not part of the sequence)
- This does not match the values given, so this statement is **false.**
Based on this evaluation, the two true statements are:
**B. The sequence is geometric.**
**D. The sequence is represented by d = 2^(n-1).**