Chapter 4.7, Problem 4P

(a)

To determine

The explicit formula to show the remaining value on the card as arithmetic sequence is to be written.

Answer to Problem 4P

The explicit formula is,

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

Explanation of Solution

Given data:

Starting value of subway pass = $\$100$

The value of the pass after one ride = $\$98.25$

The value of the pass after two rides = $\$96.5$

The value of the pass after three rides = $\$94.75$

The $nth$ term of an arithmetic sequence with first term $A\left(1\right)$ and common difference $d$ is,

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

The equation (1) represents the explicit formula to calculate the remaining values on the cards.

The value of the pass after $15$ rides is calculated as follows:

From the given data:

  $A\left(1\right)=100$

  $d=1.75$

  $n=15$

Plugin the values in equation (1)

  $\begin{array}{l}A\left(15\right)=100+\left(15-1\right)1.75\\ =124.5\end{array}$

  $A\left(15\right)=\$124.5$

(b)

To determine

The number of rides taken with $\$100$ pass is to be calculated.

Answer to Problem 4P

The number of rides is,

  $57.14\simeq 58$

Explanation of Solution

Given data:

The cost of one ride = $\$1.75\text{per}\text{ride}$

Total cost = $\text{\$100}$

Therefore, number of rides becomes,

  $\begin{array}{l}\text{=}\frac{\text{\$100}}{\$1.75/\text{ride}}\\ =57.14\simeq 58\text{rides}\end{array}$