Chapter 7.5, Problem 44PS

Solution

To state: The recursive rule for the amount of chlorine in the pool each week if you add 34 ounces of chlorine in the first week and 16 ounces every week thereafter. Each week 40 percent of the chlorine evaporates. Also tell what happens to the amount of chlorine in the pool over time.

The recursive rule is $a_1=34,a_n=0.60a_{n-1}+16$ and the amount of chlorine over time is 40 ounces.

Given information:

34 ounces of chlorine is added in the first week and 16 ounces every week thereafter.

Each week 40 percent of the chlorine evaporates.

Explanation:

34 ounces of chlorine is added in the first week and 16 ounces every week thereafter.

Each week 40 percent of the chlorine evaporates. It means only 60 percent of chlorine is left from the previous week’s chlorine.

The recursive rule for the amount of chlorine in the pool each week is:

  $a_1=34,a_n=0.60a_{n-1}+16$

  $\begin{array}{c}{a}_2=0.60a_{2-1}+16\\ =0.6a_1+16\\ =0.6\left(34\right)+16\\ =36.4\end{array}$

Similarly, $a_3=37.84$ if it is continued like this then after some steps the amount of chlorine in the pool over time is 40 ounces.