Question

a new restaurant opened five weeks ago. the owner needs to estimate the food inventory needed for the next six months. to make the estimate, she will use the sales to date from the past five weeks shown in the table and assume that the restaurant will continue to expand at the same rate. complete 1–4.

1. write an explicit definition to model the amount of each type of food needed each week.

the number of pounds of hamburger needed can be modeled by ( a_n = square ).
(simplify your answer. use integers or decimals for any numbers in the expression.)
table content (partially visible):
weeks: week 1, week 2, week 3, week 4, week 5
pounds of hamburger: 50, 75, 100, 125, 150
dozen rolls: 21, 29, 37, 45, 53
pounds of cheese: 3.1250, 4.6875, 6.2500, 7.8125, 9.3750
pounds of french fries: 83, 105, 127, 149, 171

Explanation:

Step1: Identify the sequence type

Looking at the pounds of hamburger: Week 1: 50, Week 2: 75, Week 3: 100, Week 4: 125, Week 5: 150. This is an arithmetic sequence where the common difference \( d = 75 - 50 = 25 \).

Step2: Explicit formula for arithmetic sequence

The explicit formula for an arithmetic sequence is \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference. Here, \( a_1 = 50 \) and \( d = 25 \).

Substitute \( a_1 \) and \( d \) into the formula:
\( a_n = 50 + (n - 1) \times 25 \)

Simplify the expression:
\( a_n = 50 + 25n - 25 = 25n + 25 \)

Answer:

\( 25n + 25 \)