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.
(6 points)
A. The arithmetic sequences is an=
B. After 18 days she has made a19=
bags.
C. The linear equation that represents this situation is y =
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.
(6 points)
A. The arithmetic sequences is an=
B. After 18 days she has made a19=
bags.
C. The linear equation that represents this situation is y =
Answers
GPT-4o mini
Let's address each part of the question step by step.
**A.** To write the arithmetic sequence that represents the situation, we note that Ms. Beckner has already completed 10 gift bags and will be making 5 more each day. The sequence starts at 10 and increases by 5 each day.
The \(n\)th term of an arithmetic sequence can be expressed as:
\[ a_n = a_1 + (n-1)d \]
where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
In this case:
- \(a_1 = 10\) (the first term, or the number of gift bags she starts with)
- \(d = 5\) (the number of gift bags she makes per day)
Thus, the arithmetic sequence can be expressed as:
\[ a_n = 10 + (n-1) \cdot 5 \]
or simplifying,
\[ a_n = 10 + 5n - 5 \]
\[ a_n = 5n + 5 \]
**B.** If there are 18 days until her wedding, we want to find the total number of gift bags she will have after 18 days.
At the end of 18 days, the number of gift bags made can be calculated using \(a_{19}\) (the total after 18 days of making additional bags), as day 1 corresponds to the first gift bag:
1. First, calculate the number of gift bags made during 18 days:
\[
a_{19} = 10 + (18 \times 5)
\]
\[
= 10 + 90 = 100
\]
Thus, after 18 days, she will have made \(a_{19} = 100\) bags.
**C.** To formulate the linear equation that represents this situation, we identify \(x\) as the number of days and \(y\) as the total number of gift bags.
Starting with:
- On day 0 (when she has completed the initial bags), she has 10 gift bags.
- For each additional day, she creates 5 more bags.
The linear equation can be expressed as:
\[
y = 5x + 10
\]
In summary:
**A.** The arithmetic sequence is \(a_n = 5n + 5\).
**B.** After 18 days she has made \(a_{19} = 100\) bags.
**C.** The linear equation that represents this situation is \(y = 5x + 10\).
**A.** To write the arithmetic sequence that represents the situation, we note that Ms. Beckner has already completed 10 gift bags and will be making 5 more each day. The sequence starts at 10 and increases by 5 each day.
The \(n\)th term of an arithmetic sequence can be expressed as:
\[ a_n = a_1 + (n-1)d \]
where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
In this case:
- \(a_1 = 10\) (the first term, or the number of gift bags she starts with)
- \(d = 5\) (the number of gift bags she makes per day)
Thus, the arithmetic sequence can be expressed as:
\[ a_n = 10 + (n-1) \cdot 5 \]
or simplifying,
\[ a_n = 10 + 5n - 5 \]
\[ a_n = 5n + 5 \]
**B.** If there are 18 days until her wedding, we want to find the total number of gift bags she will have after 18 days.
At the end of 18 days, the number of gift bags made can be calculated using \(a_{19}\) (the total after 18 days of making additional bags), as day 1 corresponds to the first gift bag:
1. First, calculate the number of gift bags made during 18 days:
\[
a_{19} = 10 + (18 \times 5)
\]
\[
= 10 + 90 = 100
\]
Thus, after 18 days, she will have made \(a_{19} = 100\) bags.
**C.** To formulate the linear equation that represents this situation, we identify \(x\) as the number of days and \(y\) as the total number of gift bags.
Starting with:
- On day 0 (when she has completed the initial bags), she has 10 gift bags.
- For each additional day, she creates 5 more bags.
The linear equation can be expressed as:
\[
y = 5x + 10
\]
In summary:
**A.** The arithmetic sequence is \(a_n = 5n + 5\).
**B.** After 18 days she has made \(a_{19} = 100\) bags.
**C.** The linear equation that represents this situation is \(y = 5x + 10\).