Question
Select ALL of the following that are examples of an arithmetic sequence. (1 point)
Responses
20, 14, 8, 2, −4, −10
3, 9, 27, 81, 243
You start at an elevation of 12,000 feet and climb down 1,000 feet per day.
You add $2 to a piggy bank and add triple the money you added the previous day for one month.
1, 11, 21, 25, 31, 38
Responses
20, 14, 8, 2, −4, −10
3, 9, 27, 81, 243
You start at an elevation of 12,000 feet and climb down 1,000 feet per day.
You add $2 to a piggy bank and add triple the money you added the previous day for one month.
1, 11, 21, 25, 31, 38
Answers
Answer
Given the Recursive Formula:
a1 = 5
an = an−1 +2
(4 points)
What is the common difference? Response area
What is the first term? Response area
Which term of the sequence does a4
represent? Response area
Which term does an−1
represent? Response area
a1 = 5
an = an−1 +2
(4 points)
What is the common difference? Response area
What is the first term? Response area
Which term of the sequence does a4
represent? Response area
Which term does an−1
represent? Response area
Answer
Given the explicit formula: an= 8 + 7(n − 1)
(4 points)
What is the common difference? Response area
What is the first term? Response area
What term is the formula finding if they ask for a5
? Response area
What value should replace n? Response area
(4 points)
What is the common difference? Response area
What is the first term? Response area
What term is the formula finding if they ask for a5
? Response area
What value should replace n? Response area
Answer
Given the explicit formula, find the 5th term.
an = −5 +10(n − 1)
(1 point)
Responses
45
35
-25
10
an = −5 +10(n − 1)
(1 point)
Responses
45
35
-25
10
Answer
Which function gives the correct recursive formula for the sequence?
2, 7, 12, 17, 22, 27
(1 point)
Responses
a1 = 7; an = an−1 + 2
a1 = 5; an = an −1 +2
a1 = 2; an = an −1 +5
a1 = 2; an = an −1 + 7
2, 7, 12, 17, 22, 27
(1 point)
Responses
a1 = 7; an = an−1 + 2
a1 = 5; an = an −1 +2
a1 = 2; an = an −1 +5
a1 = 2; an = an −1 + 7
Answer
A car has 121,358 miles on its odometer and accumulates an average of 255 more each week. What is the function rule that represents the total number of M miles the car will have on the odometer after w weeks? (1 point)
Responses
M(w)=121,358w+255
M(w)=121,358w+255w
M(w)=121,358+255w
M(w)=121,358+255
Responses
M(w)=121,358w+255
M(w)=121,358w+255w
M(w)=121,358+255w
M(w)=121,358+255
Answered by
GPT-4o mini
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Let's evaluate each of the provided sequences:
1. **20, 14, 8, 2, −4, −10**: The differences between consecutive terms are
- 14 - 20 = -6,
- 8 - 14 = -6,
- 2 - 8 = -6,
- -4 - 2 = -6,
- -10 - -4 = -6.
This is an arithmetic sequence with a common difference of -6.
2. **3, 9, 27, 81, 243**: The differences between consecutive terms are
- 9 - 3 = 6,
- 27 - 9 = 18,
- 81 - 27 = 54,
- 243 - 81 = 162.
The differences are not constant; this is not an arithmetic sequence.
3. **You start at an elevation of 12,000 feet and climb down 1,000 feet per day**: This represents an arithmetic sequence where you decrease your elevation by a constant amount (1,000 feet) each day.
4. **You add $2 to a piggy bank and add triple the money you added the previous day for one month**: This is not an arithmetic sequence because the amount added each day is changing based on the previous day's addition (e.g., 2, 6, 18, ...).
5. **1, 11, 21, 25, 31, 38**: The differences between consecutive terms are
- 11 - 1 = 10,
- 21 - 11 = 10,
- 25 - 21 = 4,
- 31 - 25 = 6,
- 38 - 31 = 7.
The differences are not constant; this is not an arithmetic sequence.
From this evaluation, the examples of an arithmetic sequence are:
- **20, 14, 8, 2, −4, −10**
- **You start at an elevation of 12,000 feet and climb down 1,000 feet per day.**
So the correct responses are the first and third options.
1. **20, 14, 8, 2, −4, −10**: The differences between consecutive terms are
- 14 - 20 = -6,
- 8 - 14 = -6,
- 2 - 8 = -6,
- -4 - 2 = -6,
- -10 - -4 = -6.
This is an arithmetic sequence with a common difference of -6.
2. **3, 9, 27, 81, 243**: The differences between consecutive terms are
- 9 - 3 = 6,
- 27 - 9 = 18,
- 81 - 27 = 54,
- 243 - 81 = 162.
The differences are not constant; this is not an arithmetic sequence.
3. **You start at an elevation of 12,000 feet and climb down 1,000 feet per day**: This represents an arithmetic sequence where you decrease your elevation by a constant amount (1,000 feet) each day.
4. **You add $2 to a piggy bank and add triple the money you added the previous day for one month**: This is not an arithmetic sequence because the amount added each day is changing based on the previous day's addition (e.g., 2, 6, 18, ...).
5. **1, 11, 21, 25, 31, 38**: The differences between consecutive terms are
- 11 - 1 = 10,
- 21 - 11 = 10,
- 25 - 21 = 4,
- 31 - 25 = 6,
- 38 - 31 = 7.
The differences are not constant; this is not an arithmetic sequence.
From this evaluation, the examples of an arithmetic sequence are:
- **20, 14, 8, 2, −4, −10**
- **You start at an elevation of 12,000 feet and climb down 1,000 feet per day.**
So the correct responses are the first and third options.
Answered by
GPT-4o mini
Let's analyze the recursive formula given:
1. **First term (a1)**:
- From the formula, we see that \( a_1 = 5 \).
- **Response**: **5**
2. **Common difference**:
- The recursive formula states that \( a_n = a_{n-1} + 2 \). The constant added (2) is the common difference.
- **Response**: **2**
3. **What term does \( a_4 \) represent?**:
- To find \( a_4 \), we can apply the recursive formula:
- \( a_1 = 5 \)
- \( a_2 = a_1 + 2 = 5 + 2 = 7 \)
- \( a_3 = a_2 + 2 = 7 + 2 = 9 \)
- \( a_4 = a_3 + 2 = 9 + 2 = 11 \)
- \( a_4 \) is the fourth term of the sequence.
- **Response**: **4th term**
4. **What does \( a_{n-1} \) represent?**:
- \( a_{n-1} \) is the term that comes before \( a_n \) in the sequence. Specifically, it represents the term that is one index before the \( n^{th} \) term, making it the \( (n-1)^{th} \) term.
- **Response**: **(n-1)th term**
To summarize the responses:
- Common Difference: **2**
- First Term: **5**
- \( a_4 \) represents: **4th term**
- \( a_{n-1} \) represents: **(n-1)th term**
1. **First term (a1)**:
- From the formula, we see that \( a_1 = 5 \).
- **Response**: **5**
2. **Common difference**:
- The recursive formula states that \( a_n = a_{n-1} + 2 \). The constant added (2) is the common difference.
- **Response**: **2**
3. **What term does \( a_4 \) represent?**:
- To find \( a_4 \), we can apply the recursive formula:
- \( a_1 = 5 \)
- \( a_2 = a_1 + 2 = 5 + 2 = 7 \)
- \( a_3 = a_2 + 2 = 7 + 2 = 9 \)
- \( a_4 = a_3 + 2 = 9 + 2 = 11 \)
- \( a_4 \) is the fourth term of the sequence.
- **Response**: **4th term**
4. **What does \( a_{n-1} \) represent?**:
- \( a_{n-1} \) is the term that comes before \( a_n \) in the sequence. Specifically, it represents the term that is one index before the \( n^{th} \) term, making it the \( (n-1)^{th} \) term.
- **Response**: **(n-1)th term**
To summarize the responses:
- Common Difference: **2**
- First Term: **5**
- \( a_4 \) represents: **4th term**
- \( a_{n-1} \) represents: **(n-1)th term**
Answered by
GPT-4o mini
Let's analyze the explicit formula given:
\[ a_n = 8 + 7(n - 1) \]
1. **Common Difference**:
- The explicit formula for an arithmetic sequence is generally given by:
\[
a_n = a_1 + (n-1)d
\]
where \( d \) is the common difference. In your formula, \( d \) is the coefficient of \( (n - 1) \), which is 7.
- **Response**: **7**
2. **First Term**:
- To find the first term, substitute \( n = 1 \) into the formula:
\[
a_1 = 8 + 7(1 - 1) = 8 + 7 \cdot 0 = 8
\]
- **Response**: **8**
3. **What term is the formula finding if they ask for \( a_5 \)**:
- The notation \( a_5 \) refers to the 5th term of the sequence.
- **Response**: **5th term**
4. **What value should replace \( n \)**:
- To find \( a_5 \), replace \( n \) with 5 in the formula:
\[
n = 5
\]
- **Response**: **5**
To summarize the responses:
- Common Difference: **7**
- First Term: **8**
- \( a_5 \) represents: **5th term**
- Value to replace \( n \): **5**
\[ a_n = 8 + 7(n - 1) \]
1. **Common Difference**:
- The explicit formula for an arithmetic sequence is generally given by:
\[
a_n = a_1 + (n-1)d
\]
where \( d \) is the common difference. In your formula, \( d \) is the coefficient of \( (n - 1) \), which is 7.
- **Response**: **7**
2. **First Term**:
- To find the first term, substitute \( n = 1 \) into the formula:
\[
a_1 = 8 + 7(1 - 1) = 8 + 7 \cdot 0 = 8
\]
- **Response**: **8**
3. **What term is the formula finding if they ask for \( a_5 \)**:
- The notation \( a_5 \) refers to the 5th term of the sequence.
- **Response**: **5th term**
4. **What value should replace \( n \)**:
- To find \( a_5 \), replace \( n \) with 5 in the formula:
\[
n = 5
\]
- **Response**: **5**
To summarize the responses:
- Common Difference: **7**
- First Term: **8**
- \( a_5 \) represents: **5th term**
- Value to replace \( n \): **5**
Answered by
GPT-4o mini
To find the 5th term \( a_5 \) using the explicit formula:
\[
a_n = -5 + 10(n - 1)
\]
Substituting \( n = 5 \):
\[
a_5 = -5 + 10(5 - 1)
\]
\[
= -5 + 10(4)
\]
\[
= -5 + 40
\]
\[
= 35
\]
So, the 5th term \( a_5 \) is **35**.
**Response**: **35**
\[
a_n = -5 + 10(n - 1)
\]
Substituting \( n = 5 \):
\[
a_5 = -5 + 10(5 - 1)
\]
\[
= -5 + 10(4)
\]
\[
= -5 + 40
\]
\[
= 35
\]
So, the 5th term \( a_5 \) is **35**.
**Response**: **35**
Answered by
GPT-4o mini
To find the correct recursive formula for the sequence \(2, 7, 12, 17, 22, 27\), we first identify the first term and the pattern in the sequence.
1. **First term**:
- The first term \(a_1 = 2\).
2. **Common difference**:
- The differences between consecutive terms are:
- \(7 - 2 = 5\)
- \(12 - 7 = 5\)
- \(17 - 12 = 5\)
- \(22 - 17 = 5\)
- \(27 - 22 = 5\)
- The common difference \(d = 5\).
Now, we can construct the recursive formula:
- The first term is \(a_1 = 2\).
- The recursive relationship based on the common difference is \(a_n = a_{n-1} + 5\).
Thus, the correct recursive formula is:
\[
a_1 = 2; \quad a_n = a_{n-1} + 5
\]
Now, let's compare this with the given options:
1. **a1 = 7; an = an−1 + 2** (Incorrect)
2. **a1 = 5; an = an−1 + 2** (Incorrect)
3. **a1 = 2; an = an−1 + 5** (Correct)
4. **a1 = 2; an = an−1 + 7** (Incorrect)
The correct response is:
**Response**: **a1 = 2; an = an−1 + 5**
1. **First term**:
- The first term \(a_1 = 2\).
2. **Common difference**:
- The differences between consecutive terms are:
- \(7 - 2 = 5\)
- \(12 - 7 = 5\)
- \(17 - 12 = 5\)
- \(22 - 17 = 5\)
- \(27 - 22 = 5\)
- The common difference \(d = 5\).
Now, we can construct the recursive formula:
- The first term is \(a_1 = 2\).
- The recursive relationship based on the common difference is \(a_n = a_{n-1} + 5\).
Thus, the correct recursive formula is:
\[
a_1 = 2; \quad a_n = a_{n-1} + 5
\]
Now, let's compare this with the given options:
1. **a1 = 7; an = an−1 + 2** (Incorrect)
2. **a1 = 5; an = an−1 + 2** (Incorrect)
3. **a1 = 2; an = an−1 + 5** (Correct)
4. **a1 = 2; an = an−1 + 7** (Incorrect)
The correct response is:
**Response**: **a1 = 2; an = an−1 + 5**
Answered by
GPT-4o mini
To find the appropriate function rule that represents the total number of miles \( M \) on the odometer after \( w \) weeks, we start with the initial mileage and add the amount of mileage added per week times the number of weeks.
1. **Initial mileage**: The car starts with 121,358 miles.
2. **Mileage accumulation**: The car accumulates 255 miles each week.
The function \( M(w) \) representing the total number of miles after \( w \) weeks can be expressed as:
\[
M(w) = \text{Initial mileage} + \text{(mileage per week)} \times \text{(number of weeks)}
\]
Putting this together, we have:
\[
M(w) = 121,358 + 255w
\]
Now, let's compare this with the provided options:
1. **M(w)=121,358w+255** (Incorrect)
2. **M(w)=121,358w+255w** (Incorrect)
3. **M(w)=121,358+255w** (Correct)
4. **M(w)=121,358+255** (Incorrect)
The correct response is:
**Response**: **M(w)=121,358+255w**
1. **Initial mileage**: The car starts with 121,358 miles.
2. **Mileage accumulation**: The car accumulates 255 miles each week.
The function \( M(w) \) representing the total number of miles after \( w \) weeks can be expressed as:
\[
M(w) = \text{Initial mileage} + \text{(mileage per week)} \times \text{(number of weeks)}
\]
Putting this together, we have:
\[
M(w) = 121,358 + 255w
\]
Now, let's compare this with the provided options:
1. **M(w)=121,358w+255** (Incorrect)
2. **M(w)=121,358w+255w** (Incorrect)
3. **M(w)=121,358+255w** (Correct)
4. **M(w)=121,358+255** (Incorrect)
The correct response is:
**Response**: **M(w)=121,358+255w**
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