Writing and Using Explicit Rules for Arithmetic Sequences

Let $A(n)$ be the sequence representing the amount of money left on Elishia's 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 $\$43.50.$ $$\begin{array}{c}50\stackrel{-3.25}{\to }46.75\stackrel{-3.25}{\to }43.5\stackrel{-3.25}{\to }\dots \end{array}$$ As we can see the common difference $d$ is equal to $\text{-}3.25.$

Finding our formula

We will use the generic form of an explicit formula of an arithmetic sequence. $$\begin{array}{c}A(n)=A(1)+(n-1)d\end{array}$$ In the above formula, $A(1)$ is the initial value and $d$ is the common difference. We will get our equation by substituting $A(1)=50$ and $d=\text{-}3.25.$

Finding the value of the card

Each lunch decreases the value of the card by $\$3.25.$ Therefore, to find the amount of money left on the card after buying $12$ lunches, we should subtract $12\cdot 3.25$ from the initial value $50.$ After buying $12$ lunches, the card's value will be $\$11.$