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

Answer 1

To represent this situation with an arithmetic sequence, we need to determine the starting point (the first term) and the common difference.

1. **Initial number of gift bags completed**: Ms. Beckner starts with 10 gift bags.
2. **Number of additional gift bags she plans to make each day**: She plans to make 5 more gift bags each day.

The arithmetic sequence can be expressed as follows:

- **First term (a)**: \( a_1 = 10 \) (the number of gift bags completed before starting to make more)
- **Common difference (d)**: \( d = 5 \) (the number of new gift bags made each day)

The general formula for the \( n \)-th term of an arithmetic sequence is given by:
\[
a_n = a_1 + (n-1) \cdot d
\]

So, in this case, the sequence can be expressed as:
\[
a_n = 10 + (n - 1) \cdot 5
\]

We can list the first few terms of this sequence to represent the total number of gift bags made up to several days:

- Day 0 (initial): \( a_1 = 10 \)
- Day 1: \( a_2 = 10 + 5(1) = 15 \)
- Day 2: \( a_3 = 10 + 5(2) = 20 \)
- Day 3: \( a_4 = 10 + 5(3) = 25 \)
- Day 4: \( a_5 = 10 + 5(4) = 30 \)
- Day 5: \( a_6 = 10 + 5(5) = 35 \)

Thus, the arithmetic sequence representing the total number of gift bags over the days leading up to the wedding is:
\[
10, 15, 20, 25, 30, 35, \ldots
\]

In general terms, the number of gift bags completed after \( n \) days before the wedding is given by:
\[
a_n = 10 + 5(n - 1)
\] where \( n \) starts at 1 (for the first term).