Chapter 4.7, Problem 37PPE

To determine

To find : the explicit formula to represent the amount of money left on the card as an arithmetic sequence and the value of the card after buying 12 lunches.

Answer to Problem 37PPE

Explicit formula: $A\left(n+1\right)=50-3.25n$

The value of the card after buying 12 lunches is $11

Explanation of Solution

Given sequence: We have a cafeteria card $50. After we buy lunch on Monday, its value is $46.75and after we buy lunch on Tuesday, its value is $43.50.

Calculation:

The sequence of the value of the cafeteria card is 50, 46.75, 43.50 ,….form an arithmetic sequence with first term $A\left(1\right)=50$ and common difference is $d=46.75-50=-3.25$

We know that the explicit formula for the arithmetic sequence with first term $A\left(1\right)$ and common ratio d is

  $A\left(n\right)=A\left(1\right)+\left(n-1\right)\left(d\right)$

The value of the card after buying n lunches is the ( n+ 1)th term of the sequence, so explicit formula to represent the amount of money left on the card after n lunches is

  $\begin{array}{l}A\left(n+1\right)=50+\left(n\right)\left(-3.25\right)\\ A\left(n+1\right)=50-3.25n\end{array}$

Now, the value of the card after buying 12 lunches = $A\left(13\right)=50-3.25\times 12=\$11$