Algebra and Trigonometry (6th Edition)
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ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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![### Question 12: Identifying the Function for a Given Sequence
**Problem Statement:**
Which function defines the sequence
\[ -6, -10, -14, -18, \ldots, \text{ where } f(6) = -26? \]
**Options:**
1. \( f(x) = -4x - 2 \)
2. \( f(x) = 4x - 2 \)
3. \( f(x) = -x + 32 \)
4. \( f(x) = x - 26 \)
**Explanation:**
The problem is asking to identify the correct function \( f(x) \) that generates the given sequence and satisfies the condition \( f(6) = -26 \).
#### Detailed Steps to Solve:
1. **Analyze the Sequence:**
\[ -6, -10, -14, -18, \ldots \]
Observe that each term in the sequence decreases by 4:
- From -6 to -10: \[ -10 - (-6) = -4 \]
- From -10 to -14: \[ -14 - (-10) = -4 \]
- From -14 to -18: \[ -18 - (-14) = -4 \]
This indicates a consistent decrement (common difference) of -4, suggesting that the sequence is arithmetic.
2. **Formulate a General Term for the Arithmetic Sequence:**
For an arithmetic sequence, the general term can be expressed as:
\[ a_n = a_1 + (n - 1)d \]
where \( a_1 \) is the first term and \( d \) is the common difference.
Here, \( a_1 = -6 \) and \( d = -4 \).
So,
\[ a_n = -6 + (n - 1)(-4) \]
Simplify:
\[ a_n = -6 - 4n + 4 \]
\[ a_n = -4n - 2 \]
3. **Determine the Correct Function:**
Match the general term derived with the given options:
- \( f(x) = -4x - 2 \) (Option 1) fits the general term \( a_n \) identified above.
Now, check the condition \( f(6)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff998dddf-5a1e-45ff-9e90-b0600afd3ac9%2F47257500-a94a-413d-99c7-ab7c4c39b1ac%2Fp7rt6oo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 12: Identifying the Function for a Given Sequence
**Problem Statement:**
Which function defines the sequence
\[ -6, -10, -14, -18, \ldots, \text{ where } f(6) = -26? \]
**Options:**
1. \( f(x) = -4x - 2 \)
2. \( f(x) = 4x - 2 \)
3. \( f(x) = -x + 32 \)
4. \( f(x) = x - 26 \)
**Explanation:**
The problem is asking to identify the correct function \( f(x) \) that generates the given sequence and satisfies the condition \( f(6) = -26 \).
#### Detailed Steps to Solve:
1. **Analyze the Sequence:**
\[ -6, -10, -14, -18, \ldots \]
Observe that each term in the sequence decreases by 4:
- From -6 to -10: \[ -10 - (-6) = -4 \]
- From -10 to -14: \[ -14 - (-10) = -4 \]
- From -14 to -18: \[ -18 - (-14) = -4 \]
This indicates a consistent decrement (common difference) of -4, suggesting that the sequence is arithmetic.
2. **Formulate a General Term for the Arithmetic Sequence:**
For an arithmetic sequence, the general term can be expressed as:
\[ a_n = a_1 + (n - 1)d \]
where \( a_1 \) is the first term and \( d \) is the common difference.
Here, \( a_1 = -6 \) and \( d = -4 \).
So,
\[ a_n = -6 + (n - 1)(-4) \]
Simplify:
\[ a_n = -6 - 4n + 4 \]
\[ a_n = -4n - 2 \]
3. **Determine the Correct Function:**
Match the general term derived with the given options:
- \( f(x) = -4x - 2 \) (Option 1) fits the general term \( a_n \) identified above.
Now, check the condition \( f(6)
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