Chapter 8.5, Problem 53E

a.

To determine

To write: a recursive rule for the number $a_n$ of member at the start of the nth year.

Answer to Problem 53E

  $a_{873}=58.65$ .

Explanation of Solution

Given:

The number of initial members in an online music service is $50,000$ .

And the company continuously loses 20% of the member and gain 5000 new members.

Calculation:

According to the question, the company loses 20% of its members each year, 80% remain in the company from one year to the next. Also 5,000 are acquired each year.

  $\begin{array}{l}{a}_1=50,000\\ 0.8a_{n-1}+5000\end{array}$ .

Hence, a recursive rule for the number $a_n$ of member at the start of the nth year is $0.8a_{n-1}+5000$ .

b.

To determine

To find: the number of members at the start of the 5th year.

Answer to Problem 53E

  $a_5=35,240$ .

Explanation of Solution

Given:

The number of initial members in an online music service is $50,000$ .

And the company continuously loses 20% of the member and gain 5000 new members.

Calculation:

From part (a) $a_n=0.8a_{n-1}+5000$ .

Therefore, start of the 5th year is $\begin{array}{l}{a}_5=0.8a_{5-1}+5000\\ a_5=0.8a_4+5000=35,240\end{array}$

Hence, the number of members at the start of the 5th year is $a_5=35,240$ .

c.

To determine

To describe: what happens to the number of members over time.

Answer to Problem 53E

  $25000$ members.

Explanation of Solution

Given:

The number of initial members in an online music service is $50,000$ .

And the company continuously loses 20% of the member and gain 5000 new members.

Calculation:

From part (a)

  $\begin{array}{l}{a}_1=50,000\\ a_5=35,240\\ a_n=0.8a_{n-1}+5000\end{array}$ .

Observe that the numbers of the company approach 25,000. Over time, the number stabilizes at about 25,000 people.

Hence, the number of members over time is $25000$ .