Exercise 37 Page 279

Let $A(n)$ be the arithmetic sequence representing the amount of money left on the card. At the beginning, the cafeteria card's value is $\$50.$ This means that our first term, $A(1),$ is $50.$ After the first purchase on Monday, its value is $\$46.75,$ and after the next day its value is $\$43.50.$ As we can see, the common difference $d$ is $\text{-}3.25.$

Writing an Explicit Formula

Let's recall the form of an explicit formula of an arithmetic sequence. In the above formula, $A(1)$ is the initial value and $d$ is the common difference. We will get our formula by substituting $A(1)=50$ and $d=\text{-}3.25.$ This formula gives us the terms of the arithmetic sequence formed by the amount of money left on the card.

Finding the Value of the Card

Let's review our sequence again. Notice that the $2^{\text{nd}}\text{ term}$ of the sequence is the amount of money left on the card after buying $1$ lunch, the $3^{\text{rd}}\text{ term}$ is the amount of money left on the card after buying $2$ lunches, and so on. Therefore, the $13^{\text{th}}\text{ term}$ of the sequence will give us the amount of money left on the card after buying $12$ lunches. This means that after buying $12$ lunches, the card's value will be $\$11.$