Question
Ms. Beckner makes gift bags for each of her wedding guests. She has 10 gift bags given to her by her daughter already completed. She plans to make 5 more each day until the wedding day.
A. Write the arithmetic sequence to represent this situation
B. If there are 18 days until her wedding, how many gift bags will she have to distribute to her guests?
C. What is the linear equation that represents this situation? Let x represent the number of days she makes gift bags and y represent the number of gift bags made.
A. Write the arithmetic sequence to represent this situation
B. If there are 18 days until her wedding, how many gift bags will she have to distribute to her guests?
C. What is the linear equation that represents this situation? Let x represent the number of days she makes gift bags and y represent the number of gift bags made.
Answers
GPT-4o mini
To address the questions step by step:
### A. Arithmetic Sequence
An arithmetic sequence can be written as follows:
1. The first term (initial amount of gift bags) is \( a_1 = 10 \).
2. The common difference (number of bags made per day) is \( d = 5 \).
The sequence can be represented as:
\[ a_n = a_1 + (n - 1)d \]
where \( n \) is the number of days.
So:
- For \( n = 1 \): \( a_1 = 10 \)
- For \( n = 2 \): \( a_2 = 10 + 5 = 15 \)
- For \( n = 3 \): \( a_3 = 10 + 10 = 20 \)
- And so on...
Thus, the arithmetic sequence for the total number of gift bags she will have after each day can be represented in general as:
- Day 1: 10 bags
- Day 2: 15 bags
- Day 3: 20 bags
- ...
- Day \( n \): \( 10 + 5(n - 1) \)
### B. Total Gift Bags After 18 Days
To find the total number of gift bags after 18 days, we need to calculate how many bags she will have made by then.
Using the formula for the arithmetic sequence:
- Number of gift bags made after \( n \) days is:
\[ a_n = 10 + 5(n - 1) \]
- For \( n = 18 \):
\[ a_{18} = 10 + 5(18 - 1) = 10 + 5 \times 17 = 10 + 85 = 95 \]
Therefore, after 18 days, Ms. Beckner will have **95 gift bags** in total.
### C. Linear Equation Representation
Let \( x \) represent the number of days she makes gift bags and \( y \) represent the total number of gift bags made. The linear equation can be modeled as follows:
1. The initial gift bags are 10.
2. The rate of making gift bags is 5 per day.
The equation of the line representing this situation can be written in slope-intercept form ( \( y = mx + b \) ):
- The slope \( m = 5 \) (the number of bags made per day).
- The y-intercept \( b = 10 \) (the initial amount of bags).
Thus, the linear equation is:
\[
y = 5x + 10
\]
In summary:
A. The arithmetic sequence is a representation of the total gift bags each day.
B. After 18 days, she will have 95 gift bags.
C. The linear equation representing the situation is \( y = 5x + 10 \).
### A. Arithmetic Sequence
An arithmetic sequence can be written as follows:
1. The first term (initial amount of gift bags) is \( a_1 = 10 \).
2. The common difference (number of bags made per day) is \( d = 5 \).
The sequence can be represented as:
\[ a_n = a_1 + (n - 1)d \]
where \( n \) is the number of days.
So:
- For \( n = 1 \): \( a_1 = 10 \)
- For \( n = 2 \): \( a_2 = 10 + 5 = 15 \)
- For \( n = 3 \): \( a_3 = 10 + 10 = 20 \)
- And so on...
Thus, the arithmetic sequence for the total number of gift bags she will have after each day can be represented in general as:
- Day 1: 10 bags
- Day 2: 15 bags
- Day 3: 20 bags
- ...
- Day \( n \): \( 10 + 5(n - 1) \)
### B. Total Gift Bags After 18 Days
To find the total number of gift bags after 18 days, we need to calculate how many bags she will have made by then.
Using the formula for the arithmetic sequence:
- Number of gift bags made after \( n \) days is:
\[ a_n = 10 + 5(n - 1) \]
- For \( n = 18 \):
\[ a_{18} = 10 + 5(18 - 1) = 10 + 5 \times 17 = 10 + 85 = 95 \]
Therefore, after 18 days, Ms. Beckner will have **95 gift bags** in total.
### C. Linear Equation Representation
Let \( x \) represent the number of days she makes gift bags and \( y \) represent the total number of gift bags made. The linear equation can be modeled as follows:
1. The initial gift bags are 10.
2. The rate of making gift bags is 5 per day.
The equation of the line representing this situation can be written in slope-intercept form ( \( y = mx + b \) ):
- The slope \( m = 5 \) (the number of bags made per day).
- The y-intercept \( b = 10 \) (the initial amount of bags).
Thus, the linear equation is:
\[
y = 5x + 10
\]
In summary:
A. The arithmetic sequence is a representation of the total gift bags each day.
B. After 18 days, she will have 95 gift bags.
C. The linear equation representing the situation is \( y = 5x + 10 \).