Question

which of the following is a real - world situation that could be modeled by the arithmetic sequence ( f(n)=100 + 25n ), for ( ngeq1 )?
a membership to a gym costs $25 per month plus a $100 initiation fee.
a library membership costs $100 to take out a maximum of 25 books.
a grocery store has a promotion that if you spend $100, you get a $25 discount.
an electric bill is $125 per month.

Explanation:

Step1: Analyze the arithmetic sequence formula

The formula \(f(n)=100 + 25n\) for \(n\geq1\). In an arithmetic - sequence model for a real - world cost, the constant term represents a one - time cost and the coefficient of \(n\) represents a per - unit (in this case, per - month) cost.

Step2: Analyze each option

• Option 1: Gym membership

The initiation fee of \(\$100\) is a one - time cost (similar to the constant term in the formula \(f(n)\)). The cost per month is \(\$25\). If \(n\) represents the number of months, then the total cost \(f(n)\) after \(n\) months is the sum of the initiation fee and the cost for \(n\) months. Using the formula \(f(n)=100+25n\) (where \(100\) is the initiation fee and \(25n\) is the cost for \(n\) months at \(\$25\) per month) is a valid model.

• Option 2: Library membership

The statement “a library membership costs \(\$100\) to take out a maximum of 25 books” is a one - time cost for a service (taking out books) and there is no indication of a sequence (no variable \(n\) representing the number of times a service is used in a way that would form an arithmetic sequence).

• Option 3: Grocery store promotion

If you spend \(\$100\) and get a \(\$25\) discount, it is a one - time price adjustment (\(100 - 25\)) and not a model of the form \(a+bn\) where \(n\) is a non - negative integer representing the number of times an action is repeated.

• Option 4: Electric bill

An electric bill of \(\$125\) per month is a constant (\(f(n)=125\) for all \(n\)) and not in the form \(f(n)=100 + 25n\).

Answer:

A. A membership to a gym costs \(\$25\) per month plus a \(\$100\) initiation fee.