Asked by Yes.
Which of the following statements correctly determines the rule for generating the next term in the sequence 23, 19.5, 16, 12.5, . ..?
Responses
Add the common difference of −3.5
Multiply by the common ratio of −3.5
Multiply by the common difference of −3.5
Add the common ratio of −3.5
Responses
Add the common difference of −3.5
Multiply by the common ratio of −3.5
Multiply by the common difference of −3.5
Add the common ratio of −3.5
Answers
Answered by
Yes.
Classify the following list of numbers as an arithmetic sequence, a geometric sequence, some other sequence, or not a sequence.
9, 4, −1
Responses
geometric sequence
some other sequence
not a sequence
arithmetic sequence
9, 4, −1
Responses
geometric sequence
some other sequence
not a sequence
arithmetic sequence
Answered by
Yes.
What is the common ratio in the following geometric sequence?
2, 1/2 , 1/8 , 1/32 , . . .
Responses:
1/4
4
1/2
−1.5
2, 1/2 , 1/8 , 1/32 , . . .
Responses:
1/4
4
1/2
−1.5
Answered by
Yes.
Given the first five terms of the sequence 19, 15, 11, 7, 3 . . .
19,15,11,7,3 , use subscript notation to describe a7
Responses
a7=−4
a7=−9
a7=−5
a7=−1
19,15,11,7,3 , use subscript notation to describe a7
Responses
a7=−4
a7=−9
a7=−5
a7=−1
Answered by
Yes.
Given the recursively defined formula a1=−3 , an=an−1+4 , determine which option is the correct function notation.(1 point)
Responses
f(1)=−3; f(n)=an−1+4
f(1)=−3; f(n)=f(n−1)+4
a(1)=−3; f(n)=f(n−1)+4
f(1)=−3; f(n)=f(an−1)+4
Responses
f(1)=−3; f(n)=an−1+4
f(1)=−3; f(n)=f(n−1)+4
a(1)=−3; f(n)=f(n−1)+4
f(1)=−3; f(n)=f(an−1)+4
Answered by
GPT-4o mini
To determine the rule for generating the next term in the sequence 23, 19.5, 16, 12.5, we can first find the difference between consecutive terms:
- 23 to 19.5: \( 19.5 - 23 = -3.5 \)
- 19.5 to 16: \( 16 - 19.5 = -3.5 \)
- 16 to 12.5: \( 12.5 - 16 = -3.5 \)
We see that the difference is consistently \(-3.5\).
Therefore, we can conclude that the rule for generating the next term is:
**Add the common difference of −3.5**.
So, the correct response is: **Add the common difference of −3.5**.
- 23 to 19.5: \( 19.5 - 23 = -3.5 \)
- 19.5 to 16: \( 16 - 19.5 = -3.5 \)
- 16 to 12.5: \( 12.5 - 16 = -3.5 \)
We see that the difference is consistently \(-3.5\).
Therefore, we can conclude that the rule for generating the next term is:
**Add the common difference of −3.5**.
So, the correct response is: **Add the common difference of −3.5**.
Answered by
GPT-4o mini
To classify the given list of numbers \(9, 4, -1\), we can check if it forms an arithmetic sequence or a geometric sequence.
1. **Arithmetic Sequence**: In an arithmetic sequence, the difference between consecutive terms is constant.
- From \(9\) to \(4\): \(4 - 9 = -5\)
- From \(4\) to \(-1\): \(-1 - 4 = -5\)
- Since the difference is the same (\(-5\)), this is indeed an **arithmetic sequence**.
2. **Geometric Sequence**: In a geometric sequence, the ratio between consecutive terms is constant.
- From \(9\) to \(4\): \(\frac{4}{9}\)
- From \(4\) to \(-1\): \(\frac{-1}{4}\)
- The ratios are not the same, so this is **not a geometric sequence**.
Since the sequence has a constant difference, we classify the list of numbers as an **arithmetic sequence**.
So the correct response is: **arithmetic sequence**.
1. **Arithmetic Sequence**: In an arithmetic sequence, the difference between consecutive terms is constant.
- From \(9\) to \(4\): \(4 - 9 = -5\)
- From \(4\) to \(-1\): \(-1 - 4 = -5\)
- Since the difference is the same (\(-5\)), this is indeed an **arithmetic sequence**.
2. **Geometric Sequence**: In a geometric sequence, the ratio between consecutive terms is constant.
- From \(9\) to \(4\): \(\frac{4}{9}\)
- From \(4\) to \(-1\): \(\frac{-1}{4}\)
- The ratios are not the same, so this is **not a geometric sequence**.
Since the sequence has a constant difference, we classify the list of numbers as an **arithmetic sequence**.
So the correct response is: **arithmetic sequence**.
Answered by
GPT-4o mini
To find the common ratio in a geometric sequence, we can divide any term by the preceding term. Let's calculate the common ratio using the first two terms of the given sequence \(2, \frac{1}{2}\):
\[
\text{Common Ratio (r)} = \frac{\text{Second Term}}{\text{First Term}} = \frac{\frac{1}{2}}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]
Now, let's check this ratio with the next set of terms for confirmation:
- From \(\frac{1}{2}\) to \(\frac{1}{8}\):
\[
\frac{\frac{1}{8}}{\frac{1}{2}} = \frac{1}{8} \times \frac{2}{1} = \frac{2}{8} = \frac{1}{4}
\]
- From \(\frac{1}{8}\) to \(\frac{1}{32}\):
\[
\frac{\frac{1}{32}}{\frac{1}{8}} = \frac{1}{32} \times \frac{8}{1} = \frac{8}{32} = \frac{1}{4}
\]
In each case, the common ratio \(r\) equals \(\frac{1}{4}\).
Therefore, the common ratio of the sequence is \(\frac{1}{4}\).
So the correct response is: **1/4**.
\[
\text{Common Ratio (r)} = \frac{\text{Second Term}}{\text{First Term}} = \frac{\frac{1}{2}}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]
Now, let's check this ratio with the next set of terms for confirmation:
- From \(\frac{1}{2}\) to \(\frac{1}{8}\):
\[
\frac{\frac{1}{8}}{\frac{1}{2}} = \frac{1}{8} \times \frac{2}{1} = \frac{2}{8} = \frac{1}{4}
\]
- From \(\frac{1}{8}\) to \(\frac{1}{32}\):
\[
\frac{\frac{1}{32}}{\frac{1}{8}} = \frac{1}{32} \times \frac{8}{1} = \frac{8}{32} = \frac{1}{4}
\]
In each case, the common ratio \(r\) equals \(\frac{1}{4}\).
Therefore, the common ratio of the sequence is \(\frac{1}{4}\).
So the correct response is: **1/4**.
Answered by
GPT-4o mini
To find \( a_7 \) in the sequence \( 19, 15, 11, 7, 3 \), we first need to determine the rule governing the sequence.
The sequence is decreasing, and we can calculate the difference between consecutive terms:
- \( 19 - 15 = 4 \)
- \( 15 - 11 = 4 \)
- \( 11 - 7 = 4 \)
- \( 7 - 3 = 4 \)
The common difference is \( -4 \) (we subtract \( 4 \) each time).
Using the formula for the \( n \)-th term of an arithmetic sequence:
\[
a_n = a_1 + (n - 1) \cdot d
\]
where:
- \( a_1 = 19 \) (the first term),
- \( d = -4 \) (the common difference),
- \( n = 7 \) (the term we want to find).
Now, we can substitute these values into the formula to find \( a_7 \):
\[
a_7 = 19 + (7 - 1) \cdot (-4)
\]
\[
a_7 = 19 + 6 \cdot (-4)
\]
\[
a_7 = 19 - 24
\]
\[
a_7 = -5
\]
Thus, the value for \( a_7 \) is \(-5\).
The correct response is: **a7=−5**.
The sequence is decreasing, and we can calculate the difference between consecutive terms:
- \( 19 - 15 = 4 \)
- \( 15 - 11 = 4 \)
- \( 11 - 7 = 4 \)
- \( 7 - 3 = 4 \)
The common difference is \( -4 \) (we subtract \( 4 \) each time).
Using the formula for the \( n \)-th term of an arithmetic sequence:
\[
a_n = a_1 + (n - 1) \cdot d
\]
where:
- \( a_1 = 19 \) (the first term),
- \( d = -4 \) (the common difference),
- \( n = 7 \) (the term we want to find).
Now, we can substitute these values into the formula to find \( a_7 \):
\[
a_7 = 19 + (7 - 1) \cdot (-4)
\]
\[
a_7 = 19 + 6 \cdot (-4)
\]
\[
a_7 = 19 - 24
\]
\[
a_7 = -5
\]
Thus, the value for \( a_7 \) is \(-5\).
The correct response is: **a7=−5**.
Answered by
GPT-4o mini
To interpret the recursively defined formula given as \( a_1 = -3 \) and \( a_n = a_{n-1} + 4 \), we need to express this correctly in function notation.
Let's analyze the responses:
1. **\( f(1) = -3; f(n) = a_{n-1} + 4 \)**
This is incorrect because \( f(n) \) should reference \( f(n-1) \), not \( a_{n-1} \).
2. **\( f(1) = -3; f(n) = f(n-1) + 4 \)**
This is correct. It represents the initial condition and the recursive definition correctly in terms of the function \( f \).
3. **\( a(1) = -3; f(n) = f(n-1) + 4 \)**
This is incorrect because it mixes \( a \) notation with \( f \) notation.
4. **\( f(1) = -3; f(n) = f(a_{n-1}) + 4 \)**
This is incorrect because it incorrectly refers to \( a_{n-1} \) instead of \( f(n-1) \).
Based on this analysis, the correct option that represents the recursive definition in function notation is:
**\( f(1) = -3; f(n) = f(n-1) + 4 \)**.
Let's analyze the responses:
1. **\( f(1) = -3; f(n) = a_{n-1} + 4 \)**
This is incorrect because \( f(n) \) should reference \( f(n-1) \), not \( a_{n-1} \).
2. **\( f(1) = -3; f(n) = f(n-1) + 4 \)**
This is correct. It represents the initial condition and the recursive definition correctly in terms of the function \( f \).
3. **\( a(1) = -3; f(n) = f(n-1) + 4 \)**
This is incorrect because it mixes \( a \) notation with \( f \) notation.
4. **\( f(1) = -3; f(n) = f(a_{n-1}) + 4 \)**
This is incorrect because it incorrectly refers to \( a_{n-1} \) instead of \( f(n-1) \).
Based on this analysis, the correct option that represents the recursive definition in function notation is:
**\( f(1) = -3; f(n) = f(n-1) + 4 \)**.