Question
Which of the following statements correctly determines the rule for generating the next term in the sequence 23, 19.5, 16, 12.5....(1 point)
Responses
Multiply by the common ration of -3.5
Multiply by the common ration of -3.5
Add the common difference of -3.5
Add the common difference of -3.5
Add the common ratio of -3.5
Add the common ratio of -3.5
Multiply by the common difference of -3.5
Responses
Multiply by the common ration of -3.5
Multiply by the common ration of -3.5
Add the common difference of -3.5
Add the common difference of -3.5
Add the common ratio of -3.5
Add the common ratio of -3.5
Multiply by the common difference of -3.5
Answers
Answer
Classify the following list of numbers as an arithmetic sequence, a geometric sequence, or neither.
9, 4, -1, -6
(1 point)
Responses
Neither
Neither
Geometric
Geometric
Arithmetic
Arithmetic
Not enough information given
9, 4, -1, -6
(1 point)
Responses
Neither
Neither
Geometric
Geometric
Arithmetic
Arithmetic
Not enough information given
Answer
What is the common ratio in the following geometric sequence?
2, 1/2,1/8,1/32...
(1 point)
Responses
-1.5
-1.5
1/4
1 fourth
1/2
1 half
4
2, 1/2,1/8,1/32...
(1 point)
Responses
-1.5
-1.5
1/4
1 fourth
1/2
1 half
4
Answer
Given the first five terms of the sequence 19, 15, 11, 7, 3... use the subscript notation to describe a7
.(1 point)
Responses
a7=−5
A sub 7 is equal to negative 5
a7=−9
A sub 7 is equal to negative 9
a7=−1
A sub 7 is equal to negative 1
a7=−4
.(1 point)
Responses
a7=−5
A sub 7 is equal to negative 5
a7=−9
A sub 7 is equal to negative 9
a7=−1
A sub 7 is equal to negative 1
a7=−4
Answer
Given the first five terms of the sequence 81, 27, 9, 3, 1, 13
... use the subscript notation to describe a7
.(1 point)
Responses
a7=27
A sub 7 is equal to 27
a7=1/9
A sub 7 is equal to 1 ninth
a7=3
A sub 7 is equal to 3
a7=1/27
... use the subscript notation to describe a7
.(1 point)
Responses
a7=27
A sub 7 is equal to 27
a7=1/9
A sub 7 is equal to 1 ninth
a7=3
A sub 7 is equal to 3
a7=1/27
Answered by
GPT-4o mini
The given sequence is 23, 19.5, 16, 12.5. To determine the rule for generating the next term, we can look at the differences between consecutive terms:
- From 23 to 19.5, the difference is 19.5 - 23 = -3.5.
- From 19.5 to 16, the difference is 16 - 19.5 = -3.5.
- From 16 to 12.5, the difference is 12.5 - 16 = -3.5.
Since the difference between each consecutive term is consistently -3.5, the rule for generating the next term is to **add the common difference of -3.5**.
Therefore, the correct statement is:
**Add the common difference of -3.5.**
- From 23 to 19.5, the difference is 19.5 - 23 = -3.5.
- From 19.5 to 16, the difference is 16 - 19.5 = -3.5.
- From 16 to 12.5, the difference is 12.5 - 16 = -3.5.
Since the difference between each consecutive term is consistently -3.5, the rule for generating the next term is to **add the common difference of -3.5**.
Therefore, the correct statement is:
**Add the common difference of -3.5.**
Answered by
GPT-4o mini
To classify the sequence 9, 4, -1, -6, we need to determine if it is an arithmetic sequence, a geometric sequence, or neither.
1. **Arithmetic Sequence**: In an arithmetic sequence, the difference between consecutive terms is constant.
- First difference: 4 - 9 = -5
- Second difference: -1 - 4 = -5
- Third difference: -6 - (-1) = -5
Since the difference is the same (-5) for all consecutive terms, this is an arithmetic sequence.
2. **Geometric Sequence**: In a geometric sequence, the ratio of consecutive terms is constant.
- First ratio: 4 / 9 ≈ 0.444
- Second ratio: -1 / 4 = -0.25
- Third ratio: -6 / -1 = 6
The ratios are not constant, so it is not a geometric sequence.
Since the sequence has a constant difference, it is classified as:
**Arithmetic**.
1. **Arithmetic Sequence**: In an arithmetic sequence, the difference between consecutive terms is constant.
- First difference: 4 - 9 = -5
- Second difference: -1 - 4 = -5
- Third difference: -6 - (-1) = -5
Since the difference is the same (-5) for all consecutive terms, this is an arithmetic sequence.
2. **Geometric Sequence**: In a geometric sequence, the ratio of consecutive terms is constant.
- First ratio: 4 / 9 ≈ 0.444
- Second ratio: -1 / 4 = -0.25
- Third ratio: -6 / -1 = 6
The ratios are not constant, so it is not a geometric sequence.
Since the sequence has a constant difference, it is classified as:
**Arithmetic**.
Answered by
GPT-4o mini
To find the common ratio in the geometric sequence 2, 1/2, 1/8, 1/32, we divide each term by the previous term:
1. **From 2 to 1/2**:
\[
\text{Common ratio} = \frac{1/2}{2} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}
\]
2. **From 1/2 to 1/8**:
\[
\text{Common ratio} = \frac{1/8}{1/2} = \frac{1}{8} \cdot \frac{2}{1} = \frac{2}{8} = \frac{1}{4}
\]
3. **From 1/8 to 1/32**:
\[
\text{Common ratio} = \frac{1/32}{1/8} = \frac{1}{32} \cdot \frac{8}{1} = \frac{8}{32} = \frac{1}{4}
\]
Since the common ratio is constant and equal to \( \frac{1}{4} \) for each pair of terms, the correct response is:
**1/4**.
1. **From 2 to 1/2**:
\[
\text{Common ratio} = \frac{1/2}{2} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}
\]
2. **From 1/2 to 1/8**:
\[
\text{Common ratio} = \frac{1/8}{1/2} = \frac{1}{8} \cdot \frac{2}{1} = \frac{2}{8} = \frac{1}{4}
\]
3. **From 1/8 to 1/32**:
\[
\text{Common ratio} = \frac{1/32}{1/8} = \frac{1}{32} \cdot \frac{8}{1} = \frac{8}{32} = \frac{1}{4}
\]
Since the common ratio is constant and equal to \( \frac{1}{4} \) for each pair of terms, the correct response is:
**1/4**.
Answered by
GPT-4o mini
To find \( a_7 \) in the sequence given (19, 15, 11, 7, 3...), we first need to determine the pattern of the sequence.
### Step 1: Identify the common difference
The differences between consecutive terms are as follows:
- \( 15 - 19 = -4 \)
- \( 11 - 15 = -4 \)
- \( 7 - 11 = -4 \)
- \( 3 - 7 = -4 \)
The common difference is \( -4 \).
### Step 2: Using the formula for the \( n \)-th term of an arithmetic sequence
The formula for the \( n \)-th term of an arithmetic sequence can be written as:
\[
a_n = a_1 + (n - 1) \cdot d
\]
Where:
- \( a_1 \) = first term
- \( d \) = common difference
- \( n \) = term number
In this case:
- \( a_1 = 19 \)
- \( d = -4 \)
- We want to find \( a_7 \).
### Step 3: Calculate \( a_7 \)
\[
a_7 = 19 + (7 - 1) \cdot (-4)
\]
\[
a_7 = 19 + 6 \cdot (-4)
\]
\[
a_7 = 19 - 24
\]
\[
a_7 = -5
\]
Thus, \( a_7 = -5 \).
The correct response is:
**a7 = −5** (A sub 7 is equal to negative 5).
### Step 1: Identify the common difference
The differences between consecutive terms are as follows:
- \( 15 - 19 = -4 \)
- \( 11 - 15 = -4 \)
- \( 7 - 11 = -4 \)
- \( 3 - 7 = -4 \)
The common difference is \( -4 \).
### Step 2: Using the formula for the \( n \)-th term of an arithmetic sequence
The formula for the \( n \)-th term of an arithmetic sequence can be written as:
\[
a_n = a_1 + (n - 1) \cdot d
\]
Where:
- \( a_1 \) = first term
- \( d \) = common difference
- \( n \) = term number
In this case:
- \( a_1 = 19 \)
- \( d = -4 \)
- We want to find \( a_7 \).
### Step 3: Calculate \( a_7 \)
\[
a_7 = 19 + (7 - 1) \cdot (-4)
\]
\[
a_7 = 19 + 6 \cdot (-4)
\]
\[
a_7 = 19 - 24
\]
\[
a_7 = -5
\]
Thus, \( a_7 = -5 \).
The correct response is:
**a7 = −5** (A sub 7 is equal to negative 5).
Answered by
GPT-4o mini
This sequence is \( 81, 27, 9, 3, 1, \ldots \). To find the value of \( a_7 \), we first need to determine whether the sequence is arithmetic or geometric.
### Step 1: Identify the type of sequence
1. **Check if it is geometric**:
Calculate the ratio between consecutive terms:
- \( \frac{27}{81} = \frac{1}{3} \)
- \( \frac{9}{27} = \frac{1}{3} \)
- \( \frac{3}{9} = \frac{1}{3} \)
- \( \frac{1}{3} = \frac{1}{3} \)
The common ratio is consistent: \( r = \frac{1}{3} \).
### Step 2: Use the formula for the \( n \)-th term of a geometric sequence
The formula for the \( n \)-th term of a geometric sequence is given by:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
Where:
- \( a_1 \) is the first term
- \( r \) is the common ratio
- \( n \) is the term number
### Step 3: Calculate \( a_7 \)
In this case:
- \( a_1 = 81 \)
- \( r = \frac{1}{3} \)
- We want to find \( a_7 \).
Using the formula:
\[
a_7 = 81 \cdot \left(\frac{1}{3}\right)^{7-1}
\]
\[
a_7 = 81 \cdot \left(\frac{1}{3}\right)^{6}
\]
\[
= 81 \cdot \frac{1}{729}
\]
\[
= \frac{81}{729}
\]
\[
= \frac{1}{9}
\]
Therefore, \( a_7 = \frac{1}{9} \).
The correct response is:
**a7 = 1/9** (A sub 7 is equal to 1 ninth).
### Step 1: Identify the type of sequence
1. **Check if it is geometric**:
Calculate the ratio between consecutive terms:
- \( \frac{27}{81} = \frac{1}{3} \)
- \( \frac{9}{27} = \frac{1}{3} \)
- \( \frac{3}{9} = \frac{1}{3} \)
- \( \frac{1}{3} = \frac{1}{3} \)
The common ratio is consistent: \( r = \frac{1}{3} \).
### Step 2: Use the formula for the \( n \)-th term of a geometric sequence
The formula for the \( n \)-th term of a geometric sequence is given by:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
Where:
- \( a_1 \) is the first term
- \( r \) is the common ratio
- \( n \) is the term number
### Step 3: Calculate \( a_7 \)
In this case:
- \( a_1 = 81 \)
- \( r = \frac{1}{3} \)
- We want to find \( a_7 \).
Using the formula:
\[
a_7 = 81 \cdot \left(\frac{1}{3}\right)^{7-1}
\]
\[
a_7 = 81 \cdot \left(\frac{1}{3}\right)^{6}
\]
\[
= 81 \cdot \frac{1}{729}
\]
\[
= \frac{81}{729}
\]
\[
= \frac{1}{9}
\]
Therefore, \( a_7 = \frac{1}{9} \).
The correct response is:
**a7 = 1/9** (A sub 7 is equal to 1 ninth).
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