Exercise 53 Page 448

Practice makes perfect

a Let

a_{n}

be the number of members at the beginning of the

n^{th}

year. The music service initially has

50000

members. This tells us that

a_{1} = 50000 .

Now, we will find a recursive equation for the sequence.

At the beginning of the

(n - 1)^{th}

year the number of members is

a_{n - 1} .

Each year, the company loses

20 % = 0.2

of members. The number of old members at the end of

(n - 1)^{th}

year is

(1 - 0.2) a_{n - 1} = 0.8 a_{n - 1} .

The service gains

5000

new members every year. This tells us that

a_{n} = 0.8 a_{n - 1} + 5000 .

Recursive Rule : a_{1} = 50000, a_{n} = 0.8 a_{n - 1} + 5000 for n > 1

b We are asked to find

a_{5} .

To do this we will use a spreadsheet. Let's enter

a_{1} = 50000

in cell A1 and write

= Round ((0.8) * (A1) + 5000, 2)

in cell A2. When we hit enter, we will get the following.

Next, copy cell A2, highlight cells A3 through A5, and paste.

We get

35240

in the fifth row. This tells us that

a_{5} = 35240 .

c We are asked to find what happens to the number of members over time. To do this, we will use a spreadsheet. Let's enter

a_{1} = 50000

in cell A1 and write

= Round (0.8) * (A1) + 5000

in cell A2. When we hit enter, we will get the following.

Next, copy cell A2, highlight cells A3 through A100, and paste.

Notice that the sequence stabilizes around cell A63.

After cell A62 we will keep getting the same value, $25000.02 .$ This tells us that the number of members stabilizes around $25000$ people.

Hints