Chapter 7.5, Problem 10GP

Solution

To state: What happens to the number of members over time if an online music service initially has 50000 annual members. Each year 70 percent of the members are retained each year and 5000 new members are added.

The number of members stabilizes at about 16666 members.

Given information:

An online music service initially has 50000 annual members. Each year 70 percent of the members are retained each year and 5000 new members are added.

Explanation:

Consider the given information,

An online music service initially has 50000 annual members. It means $a_1=50,000$ .

  $\begin{array}{c}\left[\begin{array}{l}\text{Members}\text{at}\\ \text{start}\text{of}\text{year}n\end{array}\right]=\left(0.7\right)\left[\begin{array}{l}\text{Members}\text{at}\\ \text{start}\text{of}\text{year}\left(n-1\right)\end{array}\right]+\left[\begin{array}{l}\text{New}m\text{embers}\\ \text{added}\end{array}\right]\\ a_n=0.7a_{n-1}+5000\end{array}$

Thus the recursive rule is $a_n=0.7a_{n-1}+5000$ .

  $\begin{array}{c}{a}_2=0.7a_{2-1}+5000\\ =0.7a_1+5000\\ =0.7\left(50000\right)+5000\\ =40000\end{array}$

Similarly, $a_3=33,000,a_4=28,100,a_5=24,670$ if it is continued like this then after many years the number of members approaches 16666.

Therefore, the number of members stabilizes at about 16666 members.