Question
Tell whether the following sequence is arithmetic. Justify your answer. If the sequence is arithmetic, write a recursive definition and an explicit formula to represent it.
negative 4 comma negative 7 comma negative 10 comma negative 13 comma ...
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Part 1
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A.
The sequence is an arithmetic sequence because the difference between every pair of consecutive terms is the same. A recursive definition is a Subscript nequals
enter your response here, and an explicit formula is a Subscript nequals
enter your response here.
(Do not simplify. Type an expression using n as the variable.)
B.
The sequence is not an arithmetic sequence.
negative 4 comma negative 7 comma negative 10 comma negative 13 comma ...
Question content area bottom
Part 1
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A.
The sequence is an arithmetic sequence because the difference between every pair of consecutive terms is the same. A recursive definition is a Subscript nequals
enter your response here, and an explicit formula is a Subscript nequals
enter your response here.
(Do not simplify. Type an expression using n as the variable.)
B.
The sequence is not an arithmetic sequence.
Answers
GPT-4o mini
To determine whether the given sequence is an arithmetic sequence, let's analyze the terms:
The sequence is:
-4, -7, -10, -13, ...
### Step 1: Check for Common Difference
In an arithmetic sequence, the difference between any two consecutive terms must be constant. Let's find the differences:
- The difference between the second term and the first term:
- \( -7 - (-4) = -7 + 4 = -3 \)
- The difference between the third term and the second term:
- \( -10 - (-7) = -10 + 7 = -3 \)
- The difference between the fourth term and the third term:
- \( -13 - (-10) = -13 + 10 = -3 \)
In all cases, the difference is \( -3 \). Since the difference is the same for every pair of consecutive terms, we conclude that the sequence is indeed an arithmetic sequence.
### Step 2: Write a Recursive Definition
The recursive definition of an arithmetic sequence is given by the formula:
- \( a_n = a_{n-1} + d \)
Where \( d \) is the common difference and \( a_1 \) is the first term.
In this case:
- The first term \( a_1 = -4 \)
- The common difference \( d = -3 \)
So, the recursive definition can be expressed as:
- \( a_1 = -4 \)
- \( a_n = a_{n-1} - 3 \) for \( n \geq 2 \)
### Step 3: Write an Explicit Formula
The explicit formula for an arithmetic sequence is given by the formula:
- \( a_n = a_1 + (n-1) \cdot d \)
Using our values:
- \( a_1 = -4 \)
- \( d = -3 \)
We can substitute these values into the formula:
- \( a_n = -4 + (n - 1)(-3) \)
- \( a_n = -4 - 3(n - 1) \)
- \( a_n = -4 - 3n + 3 \)
- \( a_n = -3n - 1 \)
### Final Answer:
Thus, the correct option is:
A.
The sequence is an arithmetic sequence because the difference between every pair of consecutive terms is the same. A recursive definition is:
- \( a_1 = -4 \)
- \( a_n = a_{n-1} - 3 \)
And the explicit formula is:
- \( a_n = -3n - 1 \)
The sequence is:
-4, -7, -10, -13, ...
### Step 1: Check for Common Difference
In an arithmetic sequence, the difference between any two consecutive terms must be constant. Let's find the differences:
- The difference between the second term and the first term:
- \( -7 - (-4) = -7 + 4 = -3 \)
- The difference between the third term and the second term:
- \( -10 - (-7) = -10 + 7 = -3 \)
- The difference between the fourth term and the third term:
- \( -13 - (-10) = -13 + 10 = -3 \)
In all cases, the difference is \( -3 \). Since the difference is the same for every pair of consecutive terms, we conclude that the sequence is indeed an arithmetic sequence.
### Step 2: Write a Recursive Definition
The recursive definition of an arithmetic sequence is given by the formula:
- \( a_n = a_{n-1} + d \)
Where \( d \) is the common difference and \( a_1 \) is the first term.
In this case:
- The first term \( a_1 = -4 \)
- The common difference \( d = -3 \)
So, the recursive definition can be expressed as:
- \( a_1 = -4 \)
- \( a_n = a_{n-1} - 3 \) for \( n \geq 2 \)
### Step 3: Write an Explicit Formula
The explicit formula for an arithmetic sequence is given by the formula:
- \( a_n = a_1 + (n-1) \cdot d \)
Using our values:
- \( a_1 = -4 \)
- \( d = -3 \)
We can substitute these values into the formula:
- \( a_n = -4 + (n - 1)(-3) \)
- \( a_n = -4 - 3(n - 1) \)
- \( a_n = -4 - 3n + 3 \)
- \( a_n = -3n - 1 \)
### Final Answer:
Thus, the correct option is:
A.
The sequence is an arithmetic sequence because the difference between every pair of consecutive terms is the same. A recursive definition is:
- \( a_1 = -4 \)
- \( a_n = a_{n-1} - 3 \)
And the explicit formula is:
- \( a_n = -3n - 1 \)