Question
Jessie wrote the arithmetic sequence of numbers -18, -13, -8, -3, 2, 7. Which rule can be used to model this sequence of numbers?
a n = -18 + 5n
a n = -23 + 5n
a n = 5 - 18n
a n = 5 - 23n
a n = -18 + 5n
a n = -23 + 5n
a n = 5 - 18n
a n = 5 - 23n
Answers
Answer
The arithmetic sequence below shows the costs, in dollars, of riding 1, 2, 3, and 4 miles (and so on) in a certain taxicab.
5.30, 7.80, 10.30, 12.80, ...
Which function represents the n^th term of this sequence, for n = 1, 2, 3, 4,... ?
f (n) = -2.50 + 7.80n
f (n) = 5.30 + 7.80n
f (n) = 5.30 + 2.50n
f (n) = 2.80 + 2.50n
5.30, 7.80, 10.30, 12.80, ...
Which function represents the n^th term of this sequence, for n = 1, 2, 3, 4,... ?
f (n) = -2.50 + 7.80n
f (n) = 5.30 + 7.80n
f (n) = 5.30 + 2.50n
f (n) = 2.80 + 2.50n
Answer
Gabriella draws a scale drawing of a baseball diamond on the coordinate plane below.The angle at each base is a right angle.
In her drawing, what is the shortest approximate unit distance from first base to home?
-4
4
4 square root2
2 square root2
In her drawing, what is the shortest approximate unit distance from first base to home?
-4
4
4 square root2
2 square root2
Answer
An environmental organization started a new online discussion group. The group had 25 members on its first day, and added 20 new members every day after that. Which recursive sequence correctly shows the number of members the environmental organization has after n days?
a 1 = 25 a n = a n-1 + 20
a 1 = 25 a n = a n+1 + 20
a 0 = 25 a n = a n+1 + 20
a 0 = 25 a n = a n+20
a 1 = 25 a n = a n-1 + 20
a 1 = 25 a n = a n+1 + 20
a 0 = 25 a n = a n+1 + 20
a 0 = 25 a n = a n+20
Answer
Adjective Agreement
Choose the nouns that agree grammatically with each of the adjectives listed. Select ALL nouns that agree with each adjective.
Remember, Spanish adjectives and nouns must agree in both gender and number .
Some questions have only one correct answer, some have more than one correct answer.
Choose your answers carefully! No partial credit will be given.
1. worker
men
the student
the teacher
my sisters
cats
my uncle
2. pretty
men
the student
the teacher
my sisters
cats
my uncle
3. studious
men
the student
the teacher
my sisters
cats
my uncle
4. intelligent
men
the student
the teacher
my sisters
cats
my uncle
5. busy
men
the student
the teacher
my sisters
cats
my uncle
6. patients
men
the student
the teacher
my sisters
cats
my uncle
Choose the nouns that agree grammatically with each of the adjectives listed. Select ALL nouns that agree with each adjective.
Remember, Spanish adjectives and nouns must agree in both gender and number .
Some questions have only one correct answer, some have more than one correct answer.
Choose your answers carefully! No partial credit will be given.
1. worker
men
the student
the teacher
my sisters
cats
my uncle
2. pretty
men
the student
the teacher
my sisters
cats
my uncle
3. studious
men
the student
the teacher
my sisters
cats
my uncle
4. intelligent
men
the student
the teacher
my sisters
cats
my uncle
5. busy
men
the student
the teacher
my sisters
cats
my uncle
6. patients
men
the student
the teacher
my sisters
cats
my uncle
Answer
what are the correct anwers for each one
Answered by
GPT-4o mini
To find the rule that models the arithmetic sequence \(-18, -13, -8, -3, 2, 7\), we need to determine the first term and the common difference.
1. **First Term (\(a_1\))**: The first term is \(-18\).
2. **Common Difference (d)**: We can find the common difference by subtracting any term from the next term:
\[
d = -13 - (-18) = 5
\]
You can verify this with other pairs:
\[
d = -8 - (-13) = 5 \quad \text{and} \quad d = -3 - (-8) = 5
\]
Thus, the common difference is \(5\).
The general formula for the \(n\)-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n-1)d
\]
Substituting \(a_1 = -18\) and \(d = 5\):
\[
a_n = -18 + (n-1) \cdot 5
\]
This can be simplified:
\[
a_n = -18 + 5n - 5 = 5n - 23
\]
Now, let's compare this with the provided options. The correct model is:
\[
a_n = 5 - 23n
\]
However, this matches with rearranging the terms of \(5n - 23\). Note the format seems unique but equivalent in solving.
Thus, the rule to model this sequence is:
\[
\boxed{a_n = -23 + 5n}
\]
1. **First Term (\(a_1\))**: The first term is \(-18\).
2. **Common Difference (d)**: We can find the common difference by subtracting any term from the next term:
\[
d = -13 - (-18) = 5
\]
You can verify this with other pairs:
\[
d = -8 - (-13) = 5 \quad \text{and} \quad d = -3 - (-8) = 5
\]
Thus, the common difference is \(5\).
The general formula for the \(n\)-th term of an arithmetic sequence can be expressed as:
\[
a_n = a_1 + (n-1)d
\]
Substituting \(a_1 = -18\) and \(d = 5\):
\[
a_n = -18 + (n-1) \cdot 5
\]
This can be simplified:
\[
a_n = -18 + 5n - 5 = 5n - 23
\]
Now, let's compare this with the provided options. The correct model is:
\[
a_n = 5 - 23n
\]
However, this matches with rearranging the terms of \(5n - 23\). Note the format seems unique but equivalent in solving.
Thus, the rule to model this sequence is:
\[
\boxed{a_n = -23 + 5n}
\]
Answered by
GPT-4o mini
To find the function that represents the \(n\)th term of the given arithmetic sequence (5.30, 7.80, 10.30, 12.80, ...), we need to determine the first term and the common difference.
1. **First Term (\(a_1\))**: The first term is \(5.30\).
2. **Common Difference (d)**: We find the common difference by subtracting any term from the next term:
\[
d = 7.80 - 5.30 = 2.50
\]
This can be verified by checking other pairs:
\[
d = 10.30 - 7.80 = 2.50 \quad \text{and} \quad d = 12.80 - 10.30 = 2.50
\]
Thus, the common difference is \(2.50\).
The general formula for the \(n\)th term of an arithmetic sequence can be expressed as:
\[
f(n) = a_1 + (n-1)d
\]
Substituting \(a_1 = 5.30\) and \(d = 2.50\):
\[
f(n) = 5.30 + (n-1) \cdot 2.50
\]
This can be simplified:
\[
f(n) = 5.30 + 2.50n - 2.50 = 2.80 + 2.50n
\]
Thus, the function representing the \(n\)th term of this sequence is:
\[
\boxed{f(n) = 2.80 + 2.50n}
\]
1. **First Term (\(a_1\))**: The first term is \(5.30\).
2. **Common Difference (d)**: We find the common difference by subtracting any term from the next term:
\[
d = 7.80 - 5.30 = 2.50
\]
This can be verified by checking other pairs:
\[
d = 10.30 - 7.80 = 2.50 \quad \text{and} \quad d = 12.80 - 10.30 = 2.50
\]
Thus, the common difference is \(2.50\).
The general formula for the \(n\)th term of an arithmetic sequence can be expressed as:
\[
f(n) = a_1 + (n-1)d
\]
Substituting \(a_1 = 5.30\) and \(d = 2.50\):
\[
f(n) = 5.30 + (n-1) \cdot 2.50
\]
This can be simplified:
\[
f(n) = 5.30 + 2.50n - 2.50 = 2.80 + 2.50n
\]
Thus, the function representing the \(n\)th term of this sequence is:
\[
\boxed{f(n) = 2.80 + 2.50n}
\]
Answered by
GPT-4o mini
To find the shortest unit distance from first base to home plate on a baseball diamond (which is essentially a square), we can analyze the coordinates of the points.
In a baseball diamond:
- **Home Plate (H)** is at \((0, 0)\)
- **First Base (1B)** is at \((1, 1)\)
To calculate the distance between these two points, we can use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates of home plate \((0, 0)\) and first base \((1, 1)\):
\[
d = \sqrt{(1 - 0)^2 + (1 - 0)^2}
\]
\[
d = \sqrt{(1)^2 + (1)^2}
\]
\[
d = \sqrt{1 + 1}
\]
\[
d = \sqrt{2}
\]
However, the values provided as options suggest the distances are scaled, so we need to consider the scaling from the points normally represented on the baseball diamond, which are usually the unit length between bases.
If we consider the scaling, the value \(2\sqrt{2}\) has the potential to represent the diagonal distance from first base back to home in a more expanded basic unit dimension rather than a basic 1 unit squared.
Thus, based on typical calculations and under the assumption from standard baseball dimensions, the most appropriate distance from first base to home plate being approximated can be represented as \(2 \sqrt{2}\).
Therefore, the answer is:
\[
\boxed{2 \sqrt{2}}
\]
In a baseball diamond:
- **Home Plate (H)** is at \((0, 0)\)
- **First Base (1B)** is at \((1, 1)\)
To calculate the distance between these two points, we can use the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates of home plate \((0, 0)\) and first base \((1, 1)\):
\[
d = \sqrt{(1 - 0)^2 + (1 - 0)^2}
\]
\[
d = \sqrt{(1)^2 + (1)^2}
\]
\[
d = \sqrt{1 + 1}
\]
\[
d = \sqrt{2}
\]
However, the values provided as options suggest the distances are scaled, so we need to consider the scaling from the points normally represented on the baseball diamond, which are usually the unit length between bases.
If we consider the scaling, the value \(2\sqrt{2}\) has the potential to represent the diagonal distance from first base back to home in a more expanded basic unit dimension rather than a basic 1 unit squared.
Thus, based on typical calculations and under the assumption from standard baseball dimensions, the most appropriate distance from first base to home plate being approximated can be represented as \(2 \sqrt{2}\).
Therefore, the answer is:
\[
\boxed{2 \sqrt{2}}
\]
Answered by
GPT-4o mini
To represent the growth of the online discussion group of the environmental organization as a recursive sequence, let's analyze the information given:
- The group starts with **25 members** on the first day.
- It **adds 20 new members** every day after the first day.
In recursive notation:
1. **Base Case**: On day 1 (which we can denote as \( n = 1 \)), the number of members is \( a_1 = 25 \).
2. **Recursive Case**: For any day after the first day, the number of members is equal to the number of members from the previous day plus the 20 new members added each day. This gives us:
\[
a_n = a_{n-1} + 20
\]
Thus, the recursive sequence should start with \( a_1 = 25 \), and the recursive formula should be \( a_n = a_{n-1} + 20 \).
Therefore, the correct recursive sequence that shows the number of members the organization has after \( n \) days is:
\[
\boxed{a_1 = 25, \quad a_n = a_{n-1} + 20}
\]
- The group starts with **25 members** on the first day.
- It **adds 20 new members** every day after the first day.
In recursive notation:
1. **Base Case**: On day 1 (which we can denote as \( n = 1 \)), the number of members is \( a_1 = 25 \).
2. **Recursive Case**: For any day after the first day, the number of members is equal to the number of members from the previous day plus the 20 new members added each day. This gives us:
\[
a_n = a_{n-1} + 20
\]
Thus, the recursive sequence should start with \( a_1 = 25 \), and the recursive formula should be \( a_n = a_{n-1} + 20 \).
Therefore, the correct recursive sequence that shows the number of members the organization has after \( n \) days is:
\[
\boxed{a_1 = 25, \quad a_n = a_{n-1} + 20}
\]
Answered by
GPT-4o mini
To select the nouns that agree grammatically with the given adjectives in Spanish, we must take into account both gender (masculine vs. feminine) and number (singular vs. plural). Here’s how they break down:
1. **worker (trabajador/trabajadora)**
- **masculine**: men (hombres), my uncle (mi tío)
- **feminine**: my sisters (mis hermanas), the teacher (la maestra)
- **plural**: cats (gatos), my sisters (mis hermanas)
- **Answers**:
- men
- the teacher
- my sisters
- cats
- my uncle
2. **pretty (hermoso/hermosa)**
- **masculine**: men (hombres), the student (el estudiante), the teacher (el maestro), my uncle (mi tío)
- **feminine**: my sisters (mis hermanas)
- **plural**: cats (gatos)
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
3. **studious (estudioso/estudiosa)**
- **masculine**: men (hombres), the student (el estudiante), the teacher (el maestro), my uncle (mi tío)
- **feminine**: my sisters (mis hermanas)
- **plural**: cats (gatos)
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
4. **intelligent (inteligente)**
- **Note**: This adjective is neutral in gender but agrees in number.
- **Singular masculine**: men (hombres), the student (el estudiante), the teacher (el maestro), my uncle (mi tío)
- **Singular feminine**: my sisters (mis hermanas)
- **Plural**: cats (gatos)
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
5. **busy (ocupado/ocupada)**
- **masculine**: men (hombres), the student (el estudiante), the teacher (el maestro), my uncle (mi tío)
- **feminine**: my sisters (mis hermanas)
- **plural**: cats (gatos)
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
6. **patients (pacientes)**
- **masculine**: men (hombres), the student (el estudiante)
- **feminine**: the teacher (la maestra), my sisters (mis hermanas)
- **plural**: cats (gatos), my uncle (mi tío)
- Since "patients" usually refers to people receiving medical attention, it is most gender-neutral but is typically associated with masculine articles in Spanish as plural:
- Therefore, "patients" can agree with plural forms:
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
### Summary of answers:
1. worker: **men**, **the teacher**, **my sisters**, **cats**, **my uncle**
2. pretty: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
3. studious: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
4. intelligent: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
5. busy: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
6. patients: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
This systematic approach ensures that we covered all aspects of gender and number for the nouns provided in relation to each adjective given.
1. **worker (trabajador/trabajadora)**
- **masculine**: men (hombres), my uncle (mi tío)
- **feminine**: my sisters (mis hermanas), the teacher (la maestra)
- **plural**: cats (gatos), my sisters (mis hermanas)
- **Answers**:
- men
- the teacher
- my sisters
- cats
- my uncle
2. **pretty (hermoso/hermosa)**
- **masculine**: men (hombres), the student (el estudiante), the teacher (el maestro), my uncle (mi tío)
- **feminine**: my sisters (mis hermanas)
- **plural**: cats (gatos)
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
3. **studious (estudioso/estudiosa)**
- **masculine**: men (hombres), the student (el estudiante), the teacher (el maestro), my uncle (mi tío)
- **feminine**: my sisters (mis hermanas)
- **plural**: cats (gatos)
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
4. **intelligent (inteligente)**
- **Note**: This adjective is neutral in gender but agrees in number.
- **Singular masculine**: men (hombres), the student (el estudiante), the teacher (el maestro), my uncle (mi tío)
- **Singular feminine**: my sisters (mis hermanas)
- **Plural**: cats (gatos)
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
5. **busy (ocupado/ocupada)**
- **masculine**: men (hombres), the student (el estudiante), the teacher (el maestro), my uncle (mi tío)
- **feminine**: my sisters (mis hermanas)
- **plural**: cats (gatos)
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
6. **patients (pacientes)**
- **masculine**: men (hombres), the student (el estudiante)
- **feminine**: the teacher (la maestra), my sisters (mis hermanas)
- **plural**: cats (gatos), my uncle (mi tío)
- Since "patients" usually refers to people receiving medical attention, it is most gender-neutral but is typically associated with masculine articles in Spanish as plural:
- Therefore, "patients" can agree with plural forms:
- **Answers**:
- men
- the student
- the teacher
- my sisters
- cats
- my uncle
### Summary of answers:
1. worker: **men**, **the teacher**, **my sisters**, **cats**, **my uncle**
2. pretty: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
3. studious: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
4. intelligent: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
5. busy: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
6. patients: **men**, **the student**, **the teacher**, **my sisters**, **cats**, **my uncle**
This systematic approach ensures that we covered all aspects of gender and number for the nouns provided in relation to each adjective given.
Answered by
GPT-4o mini
Sure! Below are the correct answers for each adjective based on gender and number agreement in Spanish:
1. **worker (trabajador/trabajadora)**
- **Correct Answers**:
- **men**
- **the teacher** (if feminine, "la maestra")
- **my sisters**
- **my uncle**
- **Applicable Nouns**: Both masculine and feminine forms are valid based on context.
2. **pretty (hermoso/hermosa)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
3. **studious (estudioso/estudiosa)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
4. **intelligent (inteligente)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
- Note: "Intelligent" is gender-neutral but agrees with both masculine and feminine nouns.
5. **busy (ocupado/ocupada)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
6. **patients (pacientes)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
### Summary of Correct Answers:
1. **worker**: men, the teacher, my sisters, my uncle
2. **pretty**: men, the student, the teacher, my sisters, my uncle
3. **studious**: men, the student, the teacher, my sisters, my uncle
4. **intelligent**: men, the student, the teacher, my sisters, my uncle
5. **busy**: men, the student, the teacher, my sisters, my uncle
6. **patients**: men, the student, the teacher, my sisters, my uncle
Let me know if you need any further clarification!
1. **worker (trabajador/trabajadora)**
- **Correct Answers**:
- **men**
- **the teacher** (if feminine, "la maestra")
- **my sisters**
- **my uncle**
- **Applicable Nouns**: Both masculine and feminine forms are valid based on context.
2. **pretty (hermoso/hermosa)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
3. **studious (estudioso/estudiosa)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
4. **intelligent (inteligente)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
- Note: "Intelligent" is gender-neutral but agrees with both masculine and feminine nouns.
5. **busy (ocupado/ocupada)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
6. **patients (pacientes)**
- **Correct Answers**:
- **men**
- **the student**
- **the teacher**
- **my sisters**
- **my uncle**
### Summary of Correct Answers:
1. **worker**: men, the teacher, my sisters, my uncle
2. **pretty**: men, the student, the teacher, my sisters, my uncle
3. **studious**: men, the student, the teacher, my sisters, my uncle
4. **intelligent**: men, the student, the teacher, my sisters, my uncle
5. **busy**: men, the student, the teacher, my sisters, my uncle
6. **patients**: men, the student, the teacher, my sisters, my uncle
Let me know if you need any further clarification!